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THE  STRENGTH  OF  THE  EARTH'S  CRUST 


JOSEPH  BARRELL 


Reprinted  for  private  circulation  from 

THE  JOURNAL  OF  GEOLOGY,  Vol.  XXII,  Nos.  1-8,  1914 

Vol.  XXIII,  Nos.  1,5,  and  6,  1915 


*     t »  * 


THE  STRENGTH  OF  THE  EARTH'S  CRUST          VQL  xxn 

PAGES 

PART  I.  Geologic  Tests  of  the  Limits  of  Strength  ....  28-  48 
PART  II.  Regional  Distribution  of  Isostatic  Compensation  .  .  145-165 
PART  III.  Influence  of  Variable  Rate  of  Isostatic  Compensation  209-236 

PART      IV.  Heterogeneity  and  Rigidity  of  the  Crust  as  Measured 

by  Departures  from  Isostasy 289-314 

PART        V.  The  Depth  of  Masses  Producing  Gravity  Anomalies 

and  Deflection  Residuals: 
Section   A,   Development   of   Criteria  for   Spheroidal 

Masses 441-468 

Section  B,  Applications  of  Criteria  to  Determine  the 

Limits  of  Depth,  Form,  and  Mass 537~555 

PART      VI.  Relations  of  Isostatic  Movements  to  a  Sphere  of  Weak- 
ness— the  Asthenosphere 655-683 

PART    VII.  Variation  of  Strength  with  Depth,  as  Shown  by  the 

Nature  of  Departures  from  Isostasy: 
Section  A,  Presentation  of  Theory 729-741 

VOL.  XXIII 

PAGES 

Section  B,  Applications  of  the  Theory 27-44 

PART  VIII.  Physical   Conditions   Controlling   the  Nature  of   the 

Lithosphere  and  Asthenosphere: 
Section  A,  Relations  between  Rigidity,  Strength,  and 

Igneous  Activity    .-* .      .      .     424-443 

Section  B,  Relations  with  Other  Fields  of  Geophysics    499-515 

Detailed  outlines  will  be  found  on  the  first  page  of  each  part,  followed  by  an 
introduction  to  and  summary  of  the  part. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST 

I.    GEOLOGIC  TESTS  OF  THE  LIMITS 
OF  STRENGTH 


JOSEPH  BARRELL 


Reprinted  for  private  circulation  from 
THE  JOURNAL  OF  GEOLOGY,  Vol.  XXII,  No.  i,  January-February  1914 


STRENGTH  OF  THE  EARTH'S  CRUST 


JOSEPH  BARRELL 
New  Haven,  Connecticut 


PREFACE 

PART  I.     GEOLOGIC  TESTS  OF  THE  LIMITS  OF  STRENGTH 

INTRODUCTORY  AND  SUMMARY 29 

MOUNTAINS  BUILT  BY  COMPRESSION  OR  IGNEOUS  ACTIVITY   ...  32 

SHIFTINGS  OF  LOAD  DUE  TO  CLIMATIC  CHANGE 33 

THE  EVIDENCE  FROM  EROSION  CYCLES 34 

THE  EVIDENCE  FROM  DEPOSITION 36 

Preliminary  Statement •.  36 

The  Deltas  of  the  Nile  and  Niger 39 

Discussion  of  Results 43 

PREFACE 

The  publication  of  a  series  of  papers  on  "  Diastrophism  and  the 
Formative  Processes"  by  T.  C.  Chamberlin  was  begun  in  the 
Journal  of  Geology  in  October,  1913.  The  second  part,  on  "Shelf- 
Seas  and  Certain  Limitations  of  Diastrophism,"  is  nearly  identical 
in  substance  with  a  portion  of  a  paper  read  by  Professor  Chamberlin 
on  August  13,  1913,  at  the  Twelfth  International  Geological  Con- 
vention at  Toronto,  Canada.  In  this  part  particularly  it  is  pointed 
out  that  the  parallel  surface  and  bottom  of  the  shelf-seas,  also  their 
occasional  extension  as  shallow  water  bodies  over  considerable  por- 
tions of  the  continent  at  certain  times,  indicate  a  relation  to  sea- 
evel  and  wave  base  rather  than  to  a  delicate  isostatic  adjustment. 
The  implications  of  this  and  other  lines  of  argument  given  by 
Chamberlin  are  toward  crustal  rigidity,  not  crustal  mobility. 

The  first  four  parts  of  the  present  article  on  "The  Strength  of 
the  Earth's  Crust"  had  been  completed  before  the  writer  read  Pro- 
fessor Chamberlin's  paper,  or  knew  that  he  was  at  work  upon  the 
subject;  but  the  conclusions  are  so  closely  in  accord  with  his, 
though  reached  by  other  lines  of  attack,  that  this  article  may  be 

28 


THE  STRENGTH  OF  THE  EARTH'S  CRUST        29 

regarded  as  a  continuation  of  the  same  subject,  an  added  contri- 
bution in  the  large  field  of  diastrophism  and  the  formative  pro- 
cesses, following  out  certain  of  its  ramifications. 

A  somewhat  general  survey  is  given  here  of  the  problem  of  the 
strength  of  the  crust,  beginning  with  the  lines  of  evidence  which 
tear  upon  it  and  following  out  to  some  degree  the  conclusions  drawn 
from  it.  It  has  in  this  way  been  cast  into  the  form  of  those  articles 
published  by  the  Journal  of  Geology  from  time  to  time,  under  the 
caption  of  "Studies  for  Students." 

PART  I.     GEOLOGIC  TESTS  OF  THE  LIMITS  OF  STRENGTH 
INTRODUCTION   AND    SUMMARY 

The  capacity  of  the  outer  crust  to  resist  vertical  stresses  is  an 
important  field  in  the  theory  of  dynamical  and  structural  geology. 
On  the  one  hand,  it  is  known  that  the  larger  segments,  those  of 
continental  and  oceanic  proportions,  rest  to  a  large  degree  in 
isostatic  equilibrium,  the  subcrust  of  the  continental  areas  being 
lighter  than  that  of  the  oceanic  areas  in  proportion  to  the  regional 
elevation.  On  the  other  hand,  the  minor  features,  those  which 
enter  into  the  composition  of  the  landscape,  are  known  to  have 
been  sculptured  by  external  forces  and  are  to  be  explained  there- 
fore as  sustained  by  reason  of  the  rigidity  of  the  crust. 

Between  these  two  extremes  in  magnitude  of  terrestrial  relief 
lie  mountain  ranges,  plateaus,  and  basins;  made  in  part  by  tangential 
forces,  modified  by  erosion  and  sedimentation.  To  what  extent 
can  these  constructional  and  destructional  forces  work  in  oppo- 
sition to  those  other  forces  which  by  producing  vertical  movement 
make  for  isostatic  equilibrium  ?  The  method  of  attack  is  from 
two  directions.  The  geologist  examines  the  structures  imposed  by 
tangential  forces,  the  mountains  built  by  igneous  extrusion,  the 
surfaces  made  by  erosion,  the  strata  consequent  upon  sedimenta- 
tion. From  them  he  may  determine  the  amount  of  strain  which 
the  crust  can  endure  before  periodic  movements  occur  in  the  'direc- 
tion of  relief  from  strain.  The  geodesist,  by  means  of  the  plumb- 
line  and  pendulum,  determines  the  subcrustal  densities  and  notes 
the  degree  to  which  these  are  balanced  against  the  relief,  pointing 


30  JOSEPH  BARRELL 

therefore  to  a  relation  of  flotational  equilibrium  within  the  solid 
earth. 

Most  geologists  in  former  years  have  utilized  but  little  the  prin- 
ciples of  isostasy,  as  may  be  seen  by  reference  to  the  standard 
manuals.  On  the  one  hand,  the  weight  of  sediments  may  be  spoken 
of  as  the  cause  of  downsinking  with  such  equal  pace  that  the  condi- 
tion of  a  shallow  sea  prevails  for  a  geologic  period,  though  perhaps 
accompanied  by  the  deposition  of  thousands  of  feet  of  sediment. 
On  the  other  hand,  and  without  argumentation  to  explain  the 
apparent  inconsistency,  the  same  geologist  may  state  that  tangen- 
tial forces  have  built  folded  mountains  miles  in  height  which  may 
be  subsequently  largely  removed  by  erosion  before  marked  vertical 
warping  of  the  crust  occurs. 

In  contrast  to  the  geologists,  certain  geodesists  have  argued 
in  recent  years  for  a  high  degree  of  isostatic  adjustment;  isostasy 
being  regarded  by  Hayford,  for  example,  as  largely  complete  in 
areas  probably  between  one  square  mile  and  one  square  degree  in 
size,  the  mean  departure  of  these  unit  areas  from  the  level  of  com- 
plete compensation  being  stated  by  him  as  ranging  from  250  to 
570  ft.  These  figures  he  does  not  regard,  however,  as  of  a  high 
order  of  accuracy,  the  latter  being  probably  the  more  reliable  of 
the  two.  He  states  that  their  significance  is  mainly  in  showing 
that  isostatic  compensation  is  nearly  perfect.  It  has  even  been 
argued  by  Button,  Willis,  and  Hayford,  as  an  outflow  of  geodetic 
studies,  that  those  vertical  movements  of  the  outer  crust  which 
tend  to  give  isostatic  equilibrium  are  the  ultimate  causes  of  the 
periodic  great  compressive  movements. 

There  is  here  between  geologists  and  geodesists  a  tendency  to 
a  fundamental  difference  of  opinion,  resulting  from  the  emphasis 
upon  one  or  the  other  of  those  opposing  forces  which  work  in  the 
outer  crust.  The  truth  must  lie  within  the  broad  zone  between 
these  two  extremes  of  theory.  To  try  to  bring  them  together  in 
harmony  is  the  problem  before  us. 

The  first  part  of  the  paper,  on  the  geologic  tests  of  the  limits  of 
strength,  opens  with  a  brief  review  of  the  lines  of  geologic  evidence 
which  may  be  used  as  tests  of  the  degree  of  resistance  or  response 
by  the  crust  to  vertical  stresses,  having  regard  to  both  area  and 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  31 

intensity.  Deltas  built  into  deep  seas  seem  best  adapted  to  give 
quantitative  measurements.  Those  of  the  Nile  and  the  Niger 
therefore  are  subjected  to  detailed  study.  They  indicate  that  the 
earth  is  competent  over  those  regions  to  sustain  stresses  due  to 
sedimentation  which  are  measured  by  the  weight  of  several  thou- 
sand feet  of  rock,  even  where  the  load  is  continuous  over  tens  of 
thousands  of  square  miles.  Whatever  response  there  may  be  is  so 
slow  that  the  deposition  is  able  to  keep  pace  with  subsidence  and 
maintain  the  load  as  a  permanent  stress  of  this  magnitude  upon  the 
crust.  By  analogy  the  conclusion  may  be  applied  to  other  parts 
of  the  earth,  and  to  those  negative  loads  created  by  the  erosion 
to  base-level  of  regions  previously  unwarped  to  an  elevation  pre- (' 
sumably  near  to  that  which  would  give  isostatic  equilibrium. 
Consequently,  also,  the  crust  should  be  able  to  bear  in  consider- 
able degree  the  folded  and  overthrusted  structures  piled  up  by  the 
tangentially  compressive  forces  which  periodically  operate  to  such 
large  degree  within  its  outer  shell.  Deeper  changes,  involving 
changes  of  density,  are  involved,  however,  in  orogenic  processes 
and  express  themselves  in  vertical  warpings  associated  with,  and 
following  after,  folding.  This  association  of  vertical  and  tangential 
forces  complicates  the  problem  of  the  crustal  strength  needed  to 
support  mountain  range?. 

The  measures  derived  from  the  study  of  deltas  are  more  in 
accord  with  those  larger  estimates  of  the  strength  of  the  crust 
obtained  by  Putnam  and  Gilbert  in  1895  fr°m  a  transcontinental 
series  of  gravity  measurements  in  which  was  developed  and  em- 
ployed for  the  first  time  the  conception  of  local  rigidity  but  regional 
isostasy.1  Their  conclusions  have  been  thought  to  be  superseded 
and  controverted,  however,  by  much  more  elaborate  and  complete 
geodetic  studies,  first  by  Hayford,  and  later  by  Hayford  and  Bowie, 
which  went  to  show  that  the  crust  was  very  much  weaker  and  in 
much  more  perfect  static  equilibrium. 

The  calculations  of  Hoskins  tended  to  show  also  that  the  crust 
within  the  zone  of  isostatic  compensation  could  not  bear  perma- 
nently loads  as  great  as  those  apparently  imposed  by  these  deltas. 
If,  however,  the  great  hydrostatic  pressures  within  the  deeper  crust 

1  Bull.  Phil.  Soc.  Wash.,  XIII  (1895),  31-75;  Jour.  GeoL,  III  (1895),  331-34. 


32  JOSEPH  BARRELL 

give  to  it  an  added  resistance  to  stress  differences  as  great  as  indi- 
cated by  the  experiments  of  Adams,  then  the  strains  imposed  by 
the  deltas  may  be  permanently  borne. 

This  confrontation  of  the  conclusions  drawn  from  various  paths 
of  approach  raises  the  problems  which  are  treated  in  the  second 
part. 

MOUNTAINS  BUILT  BY  COMPRESSION  OR  IGNEOUS  ACTIVITY 

Mountain  ranges  made  by  folding  or  extravasation  must  be 
independent  to  some  degree  from  vertical  forces,  but  these  are  not 
suitable  geologic  tests  of  the  rigidity  of  the  crust,  since  it  is  known, 
as  noted  in  the  introduction,  that  they  are  secondarily  connected 
with  diminutions  of  density  in  the  zone  of  isostatic  compensation 
and  in  many  cases  are  rejuvenated  after  partial  erosion  by  later 
up  warping. 

The  individual  mountains  or  plateau  remnants  left  standing  by 
circumdenudation,  or  piled  up  as  volcanic  cones  are  clearly  burdens 
upon  the  earth.  The  volume  which  rises  above  the  average  level 
is  a  measure  of  the  stress.  Gilbert  has  so  used  them  and  obtained 
values  ranging  from  40  to  700  cubic  miles.1  These  volumes,  how- 
ever, might  be  called  minimum  estimates,  as  may  be  seen  upon 
examination  of  their  nature. 

If  a  certain  broad  upwarping  reduces  the  vertical  stresses  to  a 
minimum  and  erosion  follows  without  further  adjustment,  it  is  the 
volume  of  the  valleys  rather  than  the  mountains  which  soon  comes 
to  measure  the  larger  possible  departures  from  equilibrium.  The 
remaining  mountains  by  their  weight  produce  local  downward 
stresses,  but  the  more  regional  stresses  are  upward  and  are  due  to 
the  breadth  of  the  field  of  erosion.  These  regional  stresses  will 
become  larger  ultimately  than  the  local  stresses  due  to  the  residual 
masses. 

Volcanic  cones  do  not  continue  to  be  built  up  until  their  base 
begins  to  sink  into  the  crust  as  fast  as  the  upward  growth  takes 
place.  On  the  contrary,  their  growth  ceases  when  the  hydrostatic 
pressure  of  the  high  column  of  lava  or  a  decadence  of  pressure  in 
the  reservoir  below  leads  finally  to  a  shifting  of  the  vents. 

1  "The  Strength  of  the  Earth's  Crust,"  Bull.  Geol.  Soc.  Am.  (1889),  I,  25. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST        33 

Regional  igneous  activity  has  poured  out  lavas  and  breccias, 
burying  previous  mountainous  topography  and  adding  thousands 
of  feet  to  the  outer  crust.  Lack  of  simultaneous  erosion,  as  in  the 
Miocene  flows  of  the  Columbia  plateau,  shows  that  subsidence 
progressed,  perhaps  with  approximately  equal  pace.  The  present 
altitude  of  the  Columbia  plateau  is  youthful,  as  shown  by  the  steep 
canyon  walls  and  undissected  interfluvial  areas.  The  initial  sub- 
sidence accompanying  igneous  outpouring  and  the  distinctly  later 
upwarping  without  compression  suggest  that  here  isostasy  has  pre- 
vailed. But  in  such  regions  the  geologic  evidence  points  toward  a 
minimum  strength  of  the  crust.  The  wide  area  of  activity,  the  nu- 
merous vents,  the  general  absence  of  localization,  all  are  suggestive 
of  widespread  fluid  rock  beneath,  magmas  which  are  probably  far 
above  the  level  where  the  accompanying  temperatures  are  normal. 
Such  conditions  would  seem  to  imply  the  impossibility  of  the  outer 
crust  carrying  over  such  regions  the  stresses  which  are  possible  in 
regions  long  free  from  igneous  activity.  More  reliance  as  maxi- 
mum measures  of  the  strength  of  the  crust  should  be  placed  there- 
fore upon  those  external  changes  which  are  entirely  independent 
in  origin  from  the  interior  of  the  earth  locally  beneath  them. 

SHEETINGS    OF   LOAD   DUE   TO   CLIMATIC   CHANGE 

Some  of  the  most  striking  examples  of  loading  and  unloading 
of  the  crust  are  those  connected  with  the  climatic  fluctuations  of 
the  Pleistocene.  The  continental  ice  sheet  formed,  advanced,  and 
retreated  rather  rapidly,  as  viewed  from  the  geologic  standpoint. 
As  it  retreated,  the  lacustrine  and  estuarine  shores  show  that  the 
land  was  rising  with  the  melting  of  the  ice.  The  upwarping  accom- 
panying deglaciation  was  limited  to  the  approximate  region  of 
maximum  glaciation  and  was  greatest  in  the  direction  where  the 
ice  was  thickest,  in  the  St.  Lawrence  valley  the  maximum  uplift 
being  more  than  600  ft.  These  relations  suggest  strongly  an  iso- 
static  response  to  the  relief  of  load.  It  is  not  known,  however,  to 
what  degree  the  previous  downwarp  compensated  for  the  burden 
of  the  continental  ice  sheet  and  what  degree  of  regional  stress  the 
crust  was  able  to  bear.  The  lack  of  close  response  is  seen  in  that 
the  up  warp  continued  as  a  residual  movement  after  the  ice  departed. 


34  JOSEPH  BARRELL 

The  movement  of  the  crust  could  not  keep  pace  with  the  climatic 
change  but  it  shows  by  means  of  these  fossil  water  planes  its  incom- 
petency  to  bear  without  at  least  partial  yielding  a  burden  as  broad 
and  as  heavy  as  the  Pleistocene  climates  placed  upon  it. 

Gilbert,  in  1889,  was  led  by  reflection  upon  the  changes  of  load 
imposed  by  the  waters  of  extinct  Lake  Bonne ville  to  use  them  as  a 
measure  of  the  strength  of  the  earth's  crust  to  resist  isostatic 
adjustments,1  and  as  previously  stated,  tested  the  conclusions 
drawn  therefrom  by  comparisons  with  the  volumes  of  mountains 
made  by  extravasation,  or  circumdenudation,  or  their  combination, 
and  of  valleys  of  erosion.  Of  Lake  Bonneville  he  states: 

Considering  the  main  body  of  Lake  Bonneville,  it  appears  from  a  study  of 
the  shorelines  that  the  removal  of  the  water  was  accompanied,  or  accompanied 
and  followed,  by  the  uprising  of  the  central  part  of  the  basin.  The  coinci- 
dence of  the  phenomena  may  have  been  fortuitous,  or  the  unloading  may  have 
been  the  cause  of  the  uprising.  Postulating  the  causal  relation,  and  assuming 
that  isostatic  equilibrium,  disturbed  by  the  removal  of  the  water,  was  restored 
by  viscous  flow  of  crust  matter,  then  it  appears  (from  observational  data) 
that  the  flow  was  not  quantitatively  sufficient  to  satisfy  the  stresses  created  by 
the  unloading.  A  stress  residium  was  left  to  be  taken  up  by  rigidity,  and  the 
measure  of  this  residuum  is  equivalent  to  the  weight  of  from  400  to  600  cubic 
miles  of  rock. 

From  these  phenomena  and  theoretic  considerations  arises  the  working 
hypothesis  that  the  measure  of  the  strength  of  the  crust  is  a  prominence  or  a 
concavity  about  600  cubic  miles  in  volume. 

THE   EVIDENCE   FROM  EROSION   CYCLES 

Erosion  base-levels  folded  and  uplifted  tracts,  leaving  for  a  time 
during  the  process  mountains  of  circumdenudation  whose  local 
stresses  have  previously  been  discussed.  The  development  of 
peneplains  implies  a  rigidity  of  the  crust  sufficient  to  prevent 
responsive  vertical  movement  until  after  the  completion  of  the 
cycle  of  denudation.  It  may  be  difficult  to  determine  the  original 
average  elevation  and  the  degree  of  progressive  uplift  pari  passu 
with  erosion  which  preceded  the  peneplanation,  but  the  fact  that 
broad  areas  become  flat  and  are  controlled  until  the  next  deforma- 
tive  movement  by  the  level  of  the  sea  suggests  that  they  cannot 

'  Bull.  Geol.  Soc.  Am.,  I  (1889),  23-27. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  35 

lie  after  erosion  in  close  isostatic  equilibrium ;  that  whatever  stress 
this  implies  can  be  carried  by  the  earth  for  long  periods  of  time. 

The  ancient  peneplains  are  now  broadly  warped  and  uplifted. 
The  rivers,  as  a  rule,  are  intrenched  in  youthful  valleys;  or  their 
seaward  courses  are  drowned  and  not  yet  reclaimed  by  delta  build- 
ing. These  features  testify  to  the  recency  of  world-wide  crustal 
unrest,  marked  chiefly  by  movements  of  a  vertical  nature;  move- 
ments which  presumably  diminished  the  vertical  stresses  in  the 
outer  portions  of  the  earth  and  has  produced  at  the  present  time,  as 
Willis  has  argued,  a  higher  degree  of  isostatic  compensation  than 
has  been  customary  through  the  long  periods  of  quiet  which  sepa- 
rate the  epochs  of  movement. 

There  are  difficulties,  however,  in  using  ancient  base-leveled 
surfaces  now  upwarped  as  measures  of  the  previous  stress.  It  is 
known  that  a  region  like  the  Colorado  plateaus  which  now  stand 
markedly  high  tended  to  lie  near  sea-level  from  the  beginning  of 
the  Paleozoic  to  the  end  of  the  Mesozoic.  Presumably  a  decrease 
of  density  within  the  zone  of  isostatic  compensation  has  taken  place 
here  during  the  Cenozoic  and  the  uplift  has  accompanied  or 
followed  the  internal  change. 

Furthermore,  if  there  are  stages  in  the  uplift,  a  considerable 
volume  of  rock  is  removed  during  each  stage,  so  that  at  no  one  time 
has  the  average  elevation  of  the  region  been  as  high  as  the  residual 
masses  might  be  thought  to  imply.  Allowing  for  these  qualifi- 
cations, however,  there  seems  no  doubt  that  the  study  of  erosion 
cycles  will  throw  light  upon  the  limits  of  stress  due  to  unloading 
which  the  crust  can  resist,  and  also  upon  progressive  changes  in 
subcrustal  densities  through  geologic  times.  This  evidence  of 
considerable  crusted  rigidity,  shown  by  freedom  from  compen- 
sating movements  during  a  cycle  of  erosion,  or  by  warpings  not  in 
sympathy  with  isostatic  stresses  during  cycles  of  crust  movements, 
has  been  pointed  out  before.  Hayford  has  sought  to  explain  it 
away  by  invoking,  first,  the  slight  crustal  cooling  which  would 
occur  in  regions  of  erosion  because  of  removal  of  the  upper  rock, 
heating  in  regions  of  deposition.  Second,  he  assumes  as  probable 
the  existence  of  a  high  coefficient  of  compressibility  sufficient  to 
make  eroded  regions  rise  in  appreciable  ratio  to  the  thickness  of 


36  JOSEPH  BARRELL 

the  load  eroded.  Third,  he  assumes  a  crustal  undertow  from 
heavy  toward  high  areas  which  would  not  only  fold  the  surface 
rocks  and  heat  them  in  the  region  of  undertow  but  restore  the 
equilibrium  of  mass  in  the  regions  of  erosion  and  deposition.1  It 
may  be  said  of  all  of  these  factors  that  when  they  are  subjected  to 
quantitative  statement  they  appear  so  trifling  as  to  fail  wholly  to 
explain  the  magnitude  and  breadth  and  periodicity  of  crust  move- 
ments. The  inadequacy  of  the  temperature  effects  has  been 
pointed  out  clearly  by  Harmon  Lewis.2  The  assumption  of  the 
high  coefficient  of  compressibility  involves  more  instead  of  less 
difficulty  for  the  high  isostasist.  The  inadequacy  of  isostatic 
undertow  to  account  for  folding  has  been  discussed  briefly  by  the 
present  writer  elsewhere.3  On  the  other  hand,  the  control  of  the 
level  of  the  earth's  surface  during  epochs  of  quiet  by  the  forces  of 
planation  and  not  by  forces  making  for  close  isostatic  adjustment 
has  been  discussed  convincingly  by  Chamberlin  in  his  present  series 
of  articles.  It  seems  clear,  then,  that  in  the  study  of  cycles  of 
erosion  and  deposition  much  may  be  determined  in  regard  to  the 
limits  of  terrestrial  rigidity.  The  subject  could  be  developed 
further,  but  it  is  preferred  to  place  the  emphasis  of  this  paper  upon 
the  more  readily  estimated  loads  produced  by  the  building  of  deltas. 

THE   EVIDENCE   FROM   DEPOSITION 

Preliminary  statement. — -The  waters  deposit  sediment  upon  the 
depressed  areas  of  the  crust.  To  what  extent  may  such  areas  be 
loaded  before  yielding  of  the  base  and  resultant  subsidence  take 
place?  The  geologic  record  makes  it  clear  that  subsidence  and 
deposition  are  necessarily  related.  It  has  been  stated  often  that 
deposition  was  the  cause  and  subsidence  the  effect,  the  two  being 
regarded  as  in  delicate  isostatic  adjustment.  But  this  is  in  reality 
an  assumption,  for  such  a  supposed  relationship  overlooks  the  extent 
to  which  subsidence  might  have  gone  forward  without  deposition 
and  ignores  the  external  load  which  may  have  been  necessary  to 

1  "The  Relations  of  Isostasy  to  Geodesy,  Geophysics,  and  Geology,"  Science, 
N.S.,  XXXIII  (1911),  199-208. 

2  "The  Theory  of  Isostasy,"  Jour.  Geol.,  XIX  (1911),  622,  623. 
s  Joseph  Barrell,  Science,  N.S.,  XXIX  (1909),  259,  260. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST        37 

perpetuate  and  add  to  a  crust  movement  initiated  by  internal 
causes.  Sedimentation  is  dependent  upon  the  rate  and  continuity 
of  subsidence  as  well  as  upon  the  rate  of  deposition.  Thus, 
although  the  sediments  give  the  most  complete  record  of  crustal 
movements,  for  the  distant  past  it  is  not  easy  to  separate  cause  and 
effect  and  ascribe  to  each  its  part.  Where  the  thickness  of  sedi- 
ments, however,  is  small,  as  over  much  of  the  continental  interior, 
the  cause  of  submergence  is  presumably  almost  wholly  independent 
of  the  local  load.  Where  the  sediments  are  thick  and  subsidence 
rapid,  as  within  the  geosynclines,  the  load  imposed  by  sedimenta- 
tion may  on  the  contrary  become  the  controlling  force.  It  is  a 
particular  phase  of  deposition,  however,  which  will  be  considered 
in  this  article,  a  study  of  the  load  imposed  upon  the  crust  by  certain 
deltas.  As  long  as  the  water  plane  lies  at  a  constant  level  the  delta 
builds  out  at  its  front.  Upon  subsidence  of  the  supporting  crust 
the  shore  retreats  inland;  less  sediment  reaches  the  now  sub- 
merged front,  and  the  delta  in  consequence  grows  chiefly  by  addi- 
tions to  the  shoreward  part  of  its  upper  surface.  The  two  methods 
of  growth  not  uncommonly  alternate  upon  the  same  delta,  showing 
the  discontinuity  of  subsidence.  In  building  outward  a  delta 
acquires  a  convex  shoreline.  This  form  is  clearly  related  to  aggra- 
dation, not  to  isostatic  uplift,  and  its  volume  is  a  measure  of  a  load 
inclined  to  further  sinking,  the  larger  rivers  tending  to  drain  toward 
and  into  the  downwarps  of  a  continent.  To  what  degree,  then, 
can  a  region  of  the  crust  which  is  possibly  already  resisting  down- 
ward strain  bear  this  added  burden  ?  A  preliminary  examination 
will  be  made  of  several  classes  of  deltas  in  order  to  choose  those 
best  adapted  to  test  this  question. 

Most  of  the  deltas  of  Eurasia  and  South  America  are  at  present 
advancing  rapidly  into  shallow  embayments  and  the  faunas  of  the 
continental  islands  show  that  the  latter  were  recently  a  part  of  the 
land.  The  physical  and  organic  evidence  thus  concur  in  showing 
that  a  very  recent  subsidence  has  taken  place.  It  is  to  be  con- 
cluded that  a  submergent  phase  in  the  Cenozoic  crustal  oscillations 
has  marked  the  short  interval  since  the  last  retreat  of  the  Pleisto- 
cene ice.  The  great  deltas  constructed  during  the  late  Tertiary 
and  in  the  Pleistocene  are  consequently  now  in  great  part  drowned. 


38  JOSEPH  BARRELL 

Their  location,  volume,  and  limits  in  most  cases  are  not  known. 
Their  modern  and  smaller  representatives,  as  they  build  out  into 
shallow  water,  do  not  greatly  increase  the  load  upon  the  crust. 
Deltas  recently  drowned  are  therefore  not  well  adapted  to  serve 
as  tests  of  the  strength  of  the  crust. 

Deltas  which  lie  in  re-entrant  angles  of  the  continents  are  also 
poorly  adapted  to  be  used  as  a  test.  Those  of  the  Indus,  the 
Ganges,  and  the  Colorado  are  illustrations.  As  they  fill  up  the 
heads  of  gulfs  and  are  without  the  typical  convex  outline,  it  is  not 
only  difficult  to  compute  their  volume  but  their  situation  is  such  as 
to  suggest  that  even  without  the  construction  of  the  delta  the 
region  might  be  far  out  of  isostatic  adjustment. 

Certain  rivers,  which  face  the  open  ocean,  such  as  the  Columbia, 
do  not  build  deltas  because  of  the  power  of  the  waves  and  currents 
which  sweep  laterally  the  fine  detritus. 

Many  rivers,  however,  build  considerable  submarine  deltas 
even  where  the  in-planing  forces  of  the  ocean  prevent  a  terrestrial 
outward  growth.  Such  submarine  deltas,  owing  probably  to  the 
power  of  the  waves  rather  than  to  recent  submergence,  are  marked 
by  convexities  in  the  bathymetric  contours  opposite  the  river 
mouths.  The  Congo,  the  Orange,  and  the  Zambesi  are  examples. 
These  hidden  deltas  which  are  built  out  into  deep  waters  cannot 
reach  more  than  a  certain  distance  from  the  shore  and  part  of  their 
detritus  is  carried  laterally  along  shore  by  the  waves,  but  never- 
theless they  possess  a  very  considerable  volume  and  the  convexity 
which  they  make  upon  the  ocean  floor  shows  to  that  degree  the 
rigidity  of  the  crust. 

The  maximum  test  is  found  where  great  rivers  have  carried 
forward  subaerial  topset  beds  of  their  deltas  over  what  was  pre- 
viously deep  ocean.  Fluviatile  construction  in  such  examples  has 
dominated  over  marine  destruction,  giving  a  convex  outline  to  the 
shore;  but  the  subaqueous  deposits  may  still  make  up  the  greater 
part  of  the  volume.  Even  in  these  cases  the  question  may  be  raised 
whether  the  deltas  have  attained  the  maximum  possible  size  per- 
mitted by  the  strength  of  the  crust.  Their  size  may,  on  the  con- 
trary, be  limited  even  here  by  the  balance  of  the  surface  agencies 
and  the  limited  time  during  which  the  river  has  dominated  over 


THE  STRENGTH  OF  THE  EARTH'S  CRUST 


39 


the  sea.  It  is  a  fair  presumption,  however,  that  the  largest  deltas 
have  reached  a  size  where  subsidence  keeps  pace  with  added 
volume. 

The  deltas  of  the  Nile  and  Niger. — Only  the  most  powerful  rivers, 
laden  with  abundant  waste  and  protected  by  their  situation  from 
the  heavier  wave  and  current  action,  can  build  deltas  of  this  last 
class  directly  into  ocean  basins.  Perhaps  the  two  best  of  the  few 
good  examples  are  those  of  the  Nile  and  the  Niger.  Both  have 
built  out  great  deltas  from  regularly  curving  shores  of  the  Atlantic 
type — the  type  where  recent  folded  mountains  do  not  mark  the 
line  between  continent  and  ocean,  the  type  where  tangential  forces 


FIG.  i. — Delta  of  the  Nile.  Scale  i:  10,000,000.  From  Andree's  Allgemeiner 
Handatlas,  vierte  Auflage. 

cannot  be  supposed  to  have  disturbed  recently  the  isostatic  balance 
of  continent  and  ocean. 

To  determine  the  areas,  depths,  and  volumes  of  the  deltas  from 
the  standpoint  of  isostasy,  a  smooth  curve,  as  shown  in  Figs,  i  and 
3,  was  continued  through  them  from  the  shore  beyond.  The  sub- 
marine contours  were  also  projected  in  dotted  lines,  giving  the  form 
of  the  bottom  as  it  presumably  would  now  be  if  no  rivers  at  these 
places  had  entered  the  sea.  The  volume  of  the  deltas  may  then  be 
determined  by  computing  the  volume  included  between  these  two 
sets  of  contour  lines. 

In  both  cases,  in  so  far  as  the  positions  of  the  hypothetical 
bottom  contours  are  open  to  doubt,  they  have  been  located  some- 
what above  a  most  probable  position,  so  as  to  tend  to  throw  the 
error  of  computation  in  the  direction  of  too  small  rather  than  too 


40  JOSEPH  BARRELL 

large  a  volume.  For  instance,  the  easterly  drift  of  the  water  facing 
the  Nile  delta  may  have  carried  considerable  mud  in  suspension 
to  beyond  the  line  assumed  here  as  its  limits.  In  consequence,  the 
hypothetical  2,ooo-meter  contour  should  be  drawn  perhaps  much 
closer  to  the  coast  of  Palestine  than  has  been  done.  Beneath  the 
Niger  delta  the  contours  lie  close  together  on  the  west  but  have 
been  drawn  as  spreading  apart  toward  the  east.  It  would  perhaps 
be  nearer  the  truth  to  project  the  steep  character  of  the  coastal 
slopes  to  the  east  of  the  Niger  delta  under  it  to  where  the  contours 
meet  the  chain  of  volcanic  island  mountains  extending  from  the 
Cameroons  out  to  sea.  This  appears  to  be  especially  probable, 
•since  Buchanan  has  shown  that  the  gentle  slopes  of  the  Guinea 
coast  even  beyond  the  limits  of  the  deltas,  and  extending  from 

2soKm.    200  ,00  °Seolevel 


FIG.  2. — Vertical  section  of  the  delta  of  the  Nile  on  A- A,  Fig.  i.  Horizontal 
scale  i:  5,000,000.  Vertical  scale  1:200,000.  Area  of  section,  295  kilometers. 

long.  2°3o'  E.  to  lat.  8°  S.,  are  mantled  throughout  by  very  soft, 
black,  oozy  mud,  characteristic  of  river  estuaries. 

All  the  way  down  the  coast  as  far  as  Loanda,  lat.  8°  S.,  the  same  gentle 
gradients  and  the  same  very  soft  river  mud  were  found.  It  appears  that  the 
land  debris  brought  down  by  the  Niger  and  Congo,  and  by  other  less  impor- 
tant rivers,  is  collected  and  concentrated  in  this  district.  The  prevailing 
current  past  the  mouth  of  the  Congo  is  a  northerly  one,  while  all  along  the 
coast  from  Cape  Palmas  to  the  Niger  an  easterly  current  sets.  These  help 
to  confine  the  drainage  matter  of  both  rivers  to  a  comparatively  small  extent 
of  littoral.  If  from  the  soundings  west  of  Cape  St.  Paul  we  compute  the 
mean  continental  slope,  we  find  that  the  5oo-fathom  line  is  at  a  mean  distance 
of  4.1  miles,  the  i,ooo-fathom  line  at  11.7  miles,  and  the  i,5oo-fathom  line 
at  a  distance  of  17  miles  from  the  loo-fathom  line.  If  it  is  assumed  that  in 
the  absence  of  the  Niger  and  the  Congo  the  continental  slope  would  be  much 
the  same  as  the  average  found  in  the  profiles  west  of  Cape  St.  Paul,  it  may  be 
concluded  that  the  excess  of  mud  forming  the  flatter  talus  along  the  coasts 
affected  by  these  rivers  is  due  to  the  mud  brought  down  by  them.1 

1  J.  Y.  Buchanan,  "On  the  Land  Slopes  Separating  Continents  and  Ocean  Basins, 
Especially  Those  on  the  West  Coast  of  Africa,"  Scottish  Geographical  Magazine,  May, 
1887,  pp.  7,  8. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST 


Buchanan  states  that  this  gentle  bottom  slope  extends  for 
1,100  miles  along  the  coast,  and  computes  the  volume  contained 
between  the  steep  gradient  presumably  once  existing  and  the  flatter 
gradient  of  the  present  bottom.  This  represents  a  deposit  of  66,- 
ooo  cubic  nautical  miles  of  detritus  due  principally  to  the  Niger 
and  the  Congo.1  This  great  volume  cannot  be  used  safely,  however, 
as  the  measure  of  a  load  upon  the  crust,  since  a  believer  in  the 
theory  of  close  isostatic  compensation  could  claim  with  some  degree 


FIG.  3. — Delta  of  the  Niger.  Scale  i :  10,000,000.  From  Andree's  Allgemeiner 
Handatlas,  vierte  Auflage. 

of  reason  that  the  initial  slope  of  the  concave  shores  of  the  Gulf  of 
Guinea  need  not  have  been  as  steep  as  the  bold  convexity  of 
Africa  to  the  west,  or  that  the  load  may  have  depressed  the 
bottom  so  as  to  have  equalized  the  pressures.  Furthermore, 
Buchanan  does  not  include  any  of  the  land  area  of  the  Niger  delta. 
The  following  estimates  will  give  the  volume  only  of  the  clearly 
constructional  part  of  the  Niger  delta,  including  both  the  land  and 

1  Op.  cit.,  p.  8  and  Fig.  3.  The  volume  stated  by  Buchanan  appears  to  be  correct 
if  the  two  profiles  have  a  common  point  taken  upon  the  shoreline.  In  his  figure,  how- 
ever, the  common  point  A  is  shown  as  upon  the  loo-fathom  contour.  From  this  error 
in  the  diagram  given  by  Buchanan  the  volume  estimated  from  the  diagram  would  be 
much  less  than  66,000  cubic  nautical  miles. 


42  JOSEPH  BARRELL 

the  sea  portion.  But  it  will  be  seen,  from  Buchanan's  statements, 
that  this  is  a  minimum  estimate  of  the  areal  load  imposed  by  the 
rivers,  for  a  more  or  less  continuous  burden  on  the  crust  would 
appear  to  stretch  for  a  thousand  miles  along  this  African  coast, 
reaching  a  maximum  unit  value,  however,  in  the  great  delta  of  the 
Niger. 

The  outer  limits  of  the  deltas  were  taken  where  the  convex 
slopes  fade  out  into  the  general  ocean  bottom. 

The  results  of  computing  the  volumes  shown  between  the  two 
sets  of  contour  lines  are  as  follows: 

TABLE  I 
DELTA  or  THE  NILE 

Area  within  i,ooo-m.  contour 71,000  sq.  km.  (27,400  sq.  mi.) 

Area  within  2,ooo-m.  contour 106,000  sq.  km.  (38,800  sq.  mi.) 

Radius  of  equivalent  circle 175  km.  (no  mi.) 

Equivalence  in  equatorial  square  degrees  8.6  sq.  degr. 

Average  thickness  within  assumed  limits 0.84  km.  (2,800  ft.) 

Equivalence  in  rock  upon  land 0.46  km.  (1,540  ft.) 

Ratio  to  76  miles  of  crust i  to  260  =  0. 0038 

Maximum  thickness 2 . 0-2 . 3  km.  (6,600-7,600  ft.) 

Equivalence  in  rock  upon  land i .  i-i .  3  km.  (3,600-4,200  ft.) 

Volume  within  assumed  limits  (extending  on 

the  east  to  somewhat  below  2,000  m.) 89,000  cu.  km.  (21,300  cu.  mi) 

Equivalence  in  rock  upon  land 50,000  cu.  km.  (11,700  cu.  mi.) 

TABLE  II 
DELTA  OF  THE  NIGER 

Area  within  the  assumed  limits 195,000  sq.  km.  (75,300  sq.  mi.) 

Radius  of  equivalent  circle 250  km.  (155  mi.) 

Equivalence  in  equatorial  square  degrees  15.8  sq.  degr. 

Average  thickness  within  assumed  limits i .  i  km.  (3,600  ft.) 

Equivalence  in  rock  upon  land 0.6  km.  (1,980  ft.) 

Ratio  to  76  miles  of  crust i  to  200=  .005 

Maximum  thickness 3.0  km.  (9,900  ft.) 

Equivalence  in  rock  upon  land i .  65  km.  (5,450  ft.) 

Volume  within  assumed  limits 217,000  cu.  km.  (52,000  cu.  mi.) 

Equivalence  in  rock  upon  land 120,000  cu.  km.  (27,000  cu.  mi.) 

The  deltas  in  their  growth  had  displaced  their  volume  of  water. 
The  added  loads  which  they  throw  upon  the  crust  are  measured  by 


THE  STRENGTH  OF  THE  EARTH'S  CRUST 


43 


subtracting  the  weight  of  the  water  from  that  of  the  sediments. 
A  specific  gavity  of  2. 67  has  been  taken  by  geodesists  as  the  aver- 
age for  the  outer  shell  of  the  earth.  The  degree  of  consolidation 
of  the  deeper  parts  of  the  deltas  is  not  known,  but  for  present  pur- 
poses the  specific  gravity  of  their  sediments  as  a  whole  may  be 
assumed  as  2.50.  This  will  be  near  the  truth  if  the  composition 
is  that  of  the  average  shale,  if  10  per  cent  of  pore  space  be  assumed 
and  this  is  wholly  filled  with  water.  The  specific  gravity  of  sea 
water  is  i .  03 ,  leaving  an  effective  specific  gravity  for  the  sediments 
of  1.47.  The  ratio  of  1.47  to  2.67  is  0.55.  The  thicknesses 
given  for  the  deltas  should  therefore  be  multipled  by  this  factor  for 
estimating  the  equivalent  burdens  of  rock  of  specific  gravity  of 
2.67  above  sea-level. 


4Ookm. 


FIG.  4. — Vertical  section  of  the  delta  of  the  Niger  on  A-A,  Fig.  3.  Horizontal 
scale  i :  5,000,000.  Vertical  scale  i :  200,000.  Area  of  the  section,  645  kilometers. 

It  is  seen  that  the  deltas  are  in  the  form  of  inclined  double 
convex  lenses.  Thicknesses  approaching  the  maximum  are  found 
over  considerable  areas  in  the  middle.  The  load  imposed  by  this 
thickness  is  equivalent  in  the  Nile  delta  to  3,600-4,200  ft.  of  rock 
above  sea-level;  in  the  Niger  delta  to  5,000-5,500  ft. 

Discussion  of  results. — The  region  of  the  southeastern  Mediter- 
ranean is  held  by  Suess  to  be  geologically  of  very  recent  origin, 
downfaulted  from  the  continent.  The  delta  of  the  Nile,  much 
smaller  than  that  of  the  Niger,  is  therefore  to  be  regarded  as  young 
and  may  be  still  increasing  in  volume. 

The  great  size  of  the  Niger  delta  suggests,  on  the  other  hand, 
that  it  may  have  reached  the  limit  permitted  by  the  strength  of 
the  crust.  Subsidence  may  now  intermittently  keep  pace  with 
deposit.  If  the  i,ooo-meter  contour  has  been  located  correctly,  as 
shown  in  Fig.  3,  it  suggests  that  such  may  be  the  case,  since  it  is 


44  JOSEPH  BARRELL 

seen  that  in  contrast  to  the  Nile  delta  the  slopes  are  much  steeper 
between  the  1,000-  to  2,ooo-meter  than  between  the  200-  to  1,000- 
meter  contours.  This  can  be  explained  by  assuming  that  the  steep 
slope  lying  below  and  beyond  a  flatter  slope  was  once  a  foreset  slope 
just  below  wave  base,  whereas  it  now  lies  at  least  800  meters  below. 
If  such  a  subsidence  has  occurred,  it  appears,  however,  to  have 
been  confined  to  within  the  limits  of  the  delta;  since  a  peripheral 
overdeepening  of  the  ocean  floor  is  not  evident.  On  the  other  hand, 
it  is  noted  by  Penck,  but  probably  too  sweepingly,  that  all  bathy- 
metric  curves  have  their  steepest  slopes  between  1,000  and  2,000 
meters  in  depth.1  Such  a  phenomenon  might  be  due  to  lateral  flow 
of  sediment  under  a  certain  depth  of  load  and  without  relation  to 
subsidence  of  the  base.  The  question  whether  the  load  of  the 
Niger  delta  is  as  great  as  the  crust  can  bear  is  therefore  an  open  one. 

The  Gulf  of  Guinea,  where  now  the  delta  is  built,  is  regarded 
by  many  geologists  as  having  originated  since  the  Middle  Mesozoic 
by  a  breaking-down  from  the  continent  of  Gondwana,  but  the 
presence  of  Middle  Cretaceous  marine  beds  skirting  much  of  the 
coast  of  West  Africa  suggests  perhaps  that  the  delta  in  its  con- 
struction does  not  go  back  of  the  Tertiary.  In  fact  it  would  seem 
possible  from  the  youthful  relief  of  the  continental  plateau  that 
the  delta  built  from  its  waste  is  of  Upper  Tertiary  and  Pleistocene 
growth. 

A  single  delta  might  happen  to  be  a  mere  veneer  of  sediment 
upon  an  originally  slightly  submerged  projecting  part  of  the  coast. 
Such  a  fortuitous  coincidence  of  unrelated  circumstances  may, 
however,  be  dismissed  as  highly  improbable  in  the  case  of  two 
great  rivers  draining  in  opposite  directions  from  the  same  continent. 
The  conclusion  that  these  deltas  are  really  externally  constructive 
features  and  measure  a  real  strain  upon  the  crust  is  strengthened 
by  noting  the  submarine  deltas  opposite  the  other  great  rivers  of 
Africa,  built  into  the  ocean,  even  though  the  waves  and  currents 
have  limited  them  by  preventing  their  subaerial  seaward  growth. 

In  the  mechanics  of  the  relation  of  the  delta  to  the  stresses  in 
the  crust  an  important  factor  is  the  nature  of  the  marginal  land. 
Shores  of  the  Pacific  type  have  great  mountain  systems  marginal 

1  Morphologic  der  Erdoberflache,  I  (1894),  146. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  45 

to  the  continents.  Parallel  to  them  the  sea  has  great  fore-deeps. 
It  appears  as  though  the  mountain  ranges  had  been  piled  too  high 
by  tangential  forces,  and,  by  virtue  of  the  partial  rigidity  of  the 
crust,  had  depressed  the  neighboring  ocean  bottoms.  Erosion  of 
the  coastal  mountains  and  deposition  of  their  waste  in  the  fore- 
deep  would  tend,  up  to  a  certain  limit,  to  equalize  the  strain  in  the 
crust.  In  that  case  it  might  happen  that,  although  the  mass  of 
the  delta  measures  a  stress,  this  might  be  opposite  in  character 
to  pre-existing  stresses,  with  the  result  that  the  strain  upon  the 
crust  beneath  the  delta  before  the  infilling  might  be  as  great  or 
greater,  but  in  an  opposite  direction.  The  greatest  remaining 
strain  within  the  sea-bottom  could  conceivably  be  an  upward 
strain  under  the  parts  of  the  fore-deep  not  filled. 

Such  relations  are  not  found  around  abyssal  slopes  of  the 
Atlantic  type.  These  are  regarded  by  many  geologists  following 
the  lead  of  Suess  as  made  by  marginal  downbreaking  of  the  con- 
tinents. They  have  but  little  or  no  relation  to  the  older  folded 
structures  and  no  excessive  deeps  parallel  to  the  continental  mar- 
gins. If  these  relations  of  the  Atlantic  and  Indian  oceans  to  the 
continents  are  rightly  interpreted  as  to  cause,  it  is  probable  that 
the  stresses  which  make  for  downsinking  extend  beyond  the  parts 
already  foundered.  The  margin  of  continents  and  ocean  basins 
are  not  likely  to  be  depressed  too  low,  but  if  remaining  out  of 
isostatic  adjustment  they  would  tend  rather  to  stand  too  high. 
There  is  no  theoretic  reason  to  believe,  therefore,  that  the  Nile  and 
Niger  deltas  have  neutralized  pre-existing  strains.  They  are  best 
regarded  as  real  and  present  burdens  sustained  by  the  rigidity  of 
the  crust. 

Whether  or  not,  however,  the  building  of  deltas  produced 
stresses  of  a  character  identical  with,  or  opposite  to,  those  previously 
existing  in  the  region,  the  stress  gradient  between  the  areas  of  the 
delta  and  the  surrounding  areas  would  be  measured  by  the  weight 
of  the  sediments,  and  this  would  tend  to  produce  differential 
flexure.  It  would  seem  to  be  a  logical  conclusion,  therefore,  from 
these  tests,  that  certain  parts  of  the  earth's  outer  crust  can  resist 
for  considerable  periods  of  time  vertical  stresses  at  least  equivalent 
to  the  weight  in  air  of  10,000-25,000  cubic  miles  of  rock  in  lenslike 


46  JOSEPH  BARRELL 

forms  spread  over  areas  of  40,000-75,000  square  miles  and  reaching 
thicknesses  in  air  over  considerable  areas  of  4,000-5,000  feet. 

The  tabulation  of  the  data  regarding  the  deltas  shows  the  area 
of  the  Niger  delta  to  be  equivalent  to  a  circle  310  miles  in  diameter 
and  that  over  this  area  the  load  of  the  delta  is  one  two-hundredths 
the  weight  of  the  crust  to  a  depth  of  76  miles,  this  being  the  depth 
of  the  zone  of  isostatic  compensation  given  by  the  latest  determi- 
nation of  Hay  ford. 

According  to  Hoskins,  in  a  calculation  made  for  Chamberlin 
and  Salisbury,1 

a  dome  corresponding  perfectly  to  the  sphericity  of  the  earth  and  formed  of 
firm  crystalline  rock  of  the  high  crushing  strength  of  25,000  pounds  to  the 
square  inch,  and  having  a  weight  of  180  pounds  to  the  cubic  foot,  would,  if 
unsupported  below,  sustain  only  ^\^  of  its  own  weight.  This  result  is  essen- 
tially independent  of  the  extent  of  the  dome,  and  also  its  thickness,  provided 
the  former  is  continental  and  the  latter  does  not  exceed  a  small  fraction  of  the 
earth's  radius. 

The  delta,  though  large,  is  so  limited  in  size  in  comparison  with 
continental  areas  that  it  would  be  somewhat  more  effectively  sup- 
ported, but  its  externally  convex  form  can  hardly  be  supposed  to 
give  it  added  domal  strength,  since  it  consists  of  more  or  less  uncon- 
solidated  material  piled  upon  a  concave  floor. 

The  theory  of  isostasy  holds  that  at  a  certain  depth  in  the  crust 
there  is  an  approach  to  equal  pressures,  the  larger  relief  of  the  sur- 
face being  balanced  in  large  part  by  subsurface  variations  in 
density.  The  larger  segments  of  the  crust  tend  to  rise  or  sink  until 
the  elevations  are  in  adjustment  to  the  density  beneath.  A  corol- 
lary of  this  theory  is  that  unbalanced  surface  loads  are  largely 
sustained  by  the  strength  of  the  crust  above  this  level  of  equal 
pressures;  in  other  words,  but  little  of  the  load  is  transmitted  to 
the  deeper  earth  below.  For  purposes  of  discussion  it  may  then  be 
assumed  that  the  load  of  the  Niger  delta  is  supported  by  the  outer 
76  miles  of  crust.  This  depth  is  one-fourth  of  the  diameter  of  the 
circle  equivalent  in  area  to  the  delta.  The  load  over  this  area,  as 
stated,  is  one  two-hundredths  of  the  weight  of  the  supporting  crust. 
Allowing  something  for  the  limited  area  of  the  delta,  it  is  seen  never- 

1  Geology,  I,  555,  1904. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST       47 

theless  to  imply  a  strength  of  the  crust  about  twice  that  assumed 
as  a  maximum  by  Hoskins  as  a  basis  for  his  calculation.  There 
are  several  contributing  factors  which  may  explain  the  disagree- 
ment between  the  figures  obtained  by  observation  of  the  deltas 
and  the  calculation  given  by  Hoskins  and  others:  First,  part  of  the 
stress  is  transmitted  laterally  to  some  extent  into  the  deeper  layers, 
but  as  the  diameter  of  the  loaded  area  is  four  times  its  depth  this 
can  be  a  partial  explanation  only  and  has,  furthermore,  been  allowed 
for.  Second,  part  of  the  stress  may  be  transmitted  into  the  deeper 
earth  below  the  yb-mile  zone  of  isostatic  compensation.  This  is 
about  equivalent  to  third,  that  the  zone  of  isostatic  compensation 
may  extend  deeper,  at  least  locally,  and  fade  out  more  after  the 
suggestion  made  by  Chamberlin.1  Fourth,  a  consideration  which 
the  writer  regards  as  most  important  is  that  the  crust  may  in  reality 
possess  greater  crushing  strength  than  the  25,000  pounds  per  inch 
postulated  by  Hoskins.  At  the  time  that  Hoskins  made  this  cal- 
culation it  seemed  that  this  figure  was  the  highest  which  could  be 
chosen,  since  it  is  higher  in  fact  than  the  crushing  strength  of  the 
average  surface  rock  when  subjected  for  even  a  short  time  to  com- 
pression in  a  testing  machine,  and  in  the  earth  the  stresses  must  be 
carried  for  indefinite  periods.  The  experiments  by  Adams2  have 
shown,  however,  that  under  the  conditions  of  cubic  compression 
which  exist  in  the  earth  the  rocks  are  capable  of  sustaining  for 
indefinite  times  far  higher  stress  differences  than  they  could  bear 
even  for  a  short  time  when  subjected  to  stress  in  one  direction  only, 
as  at  the  surface  of  the  earth.  These  experiments  showed  that: 

At  ordinary  temperatures  but  under  the  conditions  of  hydrostatic  pressure 
or  cubic  compression  which  exist  within  the  earth's  crust,  granite  will  sustain 
a  load  of  nearly  100  tons  to  the  square  inch,  that  is  to  say,  a  load  rather  more 
than  seven  times  as  great  as  that  which  will  crush  it  at  the  surface  of  the  earth 
under  the  conditions  of  the  usual  laboratory  test. 

Under  the  conditions  of  pressure  and  temperature  which  are  believed  to 
obtain  within  the  earth's  crust  empty  cavities  may  exist  in  granite  to  a  depth 
of  at  least  n  miles.3 

«. 

1  Jour.  Geol.,  XV  (1907),  76. 

2  "An  Experimental  Contribution  to  the  Question  of  the  Depth  of  the  Zone  of 
Flow  in  the  Earth's  Crust,"  Jour.  GeoL,  XX  (19^2),  97-118. 

*0p.  cit.,  p.  117. 

'  I 


48  JOSEPH  BARRELL 

It  appears  then  that,  even  allowing  for  the  great  increase  in  tem- 
perature within  the  earth's  crust  at  depths  greater  than  can  be 
reached  by  the  limitations  of  experiment,  the  demands  made  upon 
the  strength  of  the  crust  by  the  load  of  the  Niger  delta  are  not 
greater  than  can  be  explained  by  the  theory  of  the  mechanics  of 
materials  as  now  understood.  This  theory  rests,  however,  even 
after  Adams'  experiments,  upon  only  a  limited  range  of  laboratory 
observation,  and  extending  over  but  limited  periods  only,  thus 
demanding  extrapolation  both  of  stress  and  of  time  when  applied 
to  the  whole  thickness  of  the  outer  crust  and  over  hundreds  of 
thousands  of  years.  Therefore  the  study  of  the  direct  evidence 
supplied  by  geologic  observation  is  more  convincing  in  regard  to 
the  limits  of  crustal  strength. 

These  deltas  point  toward  a  measure  of  crustal  rigidity  capable 
of  sustaining  to  a  large  degree  the  downward  strains  due  to  the 
piling-up  and  overthrusting  of  mountains  built  by  tangential 
forces,  or  those  resulting  from  the  load  of  sediments  in  areas  of 
deposition,  or  those  upward  strains  produced  by  the  erosion  of 
plateaus  previously  uplifted  toward  isostatic  equilibrium.  A 
final  conclusion  must,  however,  await  a  further  discussion  in  the 

later  parts. 

[To  be  continued} 


THE  STRENGTH  OF  THE  EARTH'S  CRUST 


JOSEPH  BARRELL 
New  Haven,  Connecticut 


PART  II.    REGIONAL  DISTRIBUTION  OF  ISOSTATIC 
COMPENSATION 

INTRODUCTION  AND  SUMMARY 145 

GEODETIC  MEASUREMENTS  OF  ISOSTASY  BY  HAYFORD  AND  BOWIE  .  149 
Hayford's  Conclusions  from  Deflections  of  the  Vertical  .  .  .  149 
Hayford  and  Bowie  on  Variations  of  Gravity 152 

REGIONAL  VERSUS  LOCAL  DISTRIBUTION  OF  COMPENSATION     .      .      .  156 

Conclusions  on  This  Topic  by  Hayford  and  Bowie  .       .      .       .  156 

Review  and  Analysis  of  the  Evidence 157 

The  Test  by  Adjacent  Stations  at  Different  Elevations  .       .      .  160 

The  Test  by  Areas  of  Grouped  Residuals 162 

INTRODUCTION  AND   SUMMARY 

The  strength  of  the  crust  has  been  tested  in  the  first  part  of  this 
paper  by  those  geologic  changes  which  alter  the  surface  of  the 
earth,  but  not  the  density  of  its  interior.  If  these  changes  in  load 
initiate  rather  than  merely  coincide  with  vertical  movements  which 
serve  to  diminish  the  stress,  they  are  thereby  shown  to  be  greater 
than  the  earth  can  permanently  endure.  If,  on  the  other  hand,  the 
constructional  forms  persist,  as  in  the  two  great  deltas  studied,  then 
the  movements  which  may  exist  in  the  crust  due  to  those  loads 
must  be  slower  at  least  than  the  process  of  surface  construction. 
Such  loads  consequently,  unless  counterbalanced  by  some  factor 
not  apparent,  are  within  the  limits  of  crustal  strength. 

But  surface  changes  and  the  loads  implied  can  be  measured  only 
in  special  cases.  The  previous  attitude  of  the  crust  and  the  degree 
and  direction  of  strain  then  existing  in  it  are  complicating  factors 
which  it  is  difficult  quantitatively  to  evaluate.  For  these  reasons 
the  evidence  yielded  by  geodetic  investigation  promises,  in  the  end, 
more  general  and  more  accurate  results. 

145 


146  JOSEPH  BARRELL 

It  is  an  important  conclusion,  established  by  geodetic  evidence, 
that  the  ocean  basins  are  underlain  by  heavier  matter  than  that 
beneath  the  continental  platforms;  the  tendency  through  geologic 
time  for  the  continents  to  rise  relatively  to  the  oceans  may  be 
correlated  with  this  difference  in  density  and  the  lightening  of  the 
land  areas  by  the  progressive  erosion  of  the  land  surfaces.  It  is 
believed  that  the  rejuvenative  movements  are  in  the  direction  of 
isostatic  equilibrium.  Fortunately  for  land-dwelling  vertebrates, 
the  crust  is  too  weak  for  readjustment  to  be  deferred  until  after  the 
erosion  of  the  lands,  begun  by  the  subaerial  forces,  shall  have  been 
completed  by  the  sea. 

But  the  power  of  geodetic  research  does  not  cease  with  the 
establishment  of  this  cause  of  the  maintenance  of  the  differential 
relief  between  land  surface  and  ocean  floors.  Beneath  the  surface 
of  the  continents  it  reveals  heterogeneities  of  density  and  measures 
them  against  the  more  or  less  local  relief  above.  To  the  extent  to 
which  areas  of  lighter  or  denser  matter  do  not  correspond  to  pro- 
portionately higher  or  lower  relief,  real  strains  either  upward  or 
downward  are  shown  to  exist  through  the  crust.  Over  areas  of 
plains  which  have  not  suffered  much  change  for  geologic  ages, 
geodesy  may  thus  reveal  the  existence  of  large  crustal  strain.  On  the 
contrary,  in  regions  of  mountainous  relief,  although  the  individual 
mountains  are  sustained  by  rigidity  and  bring  local  strains  upon  the 
supporting  basement,  geodetic  study  may  show  that  there  is  close 
regional  compensation  of  density  balanced  against  relief,  obliterat- 
ing with  depth  the  stress  differences  due  to  topography,  l&ese 
methods  of  research  are  thus  capable  of  attacking  the  problem  of 
the  amount  and  direction  of  vertical  strain  existing  in  the  crust 
under  any  part  of  the  land  surface  and,  to  a  lesser  degree  of  accuracy, 
the  crust  beneath  the  sea.  The  breadth  of  the  individual  areas 
which  depart  from  equilibrium  in  one  direction  may  constitute  also 
a  vital  part  of  the  problem. 

But  although  these  are  fields  of  research  open  to  the  geodesist, 
they  are  cultivated  with  much  labor.  The  position  of  many  sta- 
tions on  the  surface  of  the  earth  must  be  determined  by  astronomic 
observations  to  within  a  fraction  of  a  second  of  arc.  Then  a 
triangulation  network,  continent-wide,  ties  these  together  and  shows 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  147 

at  each  station,  after  allowing  for  the  small  errors  of  observation, 
what  are  the  deflections  of  the  vertical  produced  by  the  variations 
of  relief  and  density.  But  this  deflection  for  each  station  is  the  net 
result  of  all  the  relief  from  mean  level  and  all  the  subsurface 
departures  from  the  densities  necessary  to.  sustain  that  relief  for 
distances  of  hundreds  and,  to  a  diminishing  extent,  even  thousands 
of  miles.  The  problem  is  made  more  soluble,  however,  by  another 
and  independent  mode  of  attack.  Observations  on  the  intensity  of 
gravity,  when  corrected  for  latitude,  for  elevation,  for  the  sur- 
rounding relief  and  the  density  theoretically  needed  to  sustain  that 
relief,  show  the  vertical  component  of  those  outstanding  forces 
whose  horizontal  component  was  measured  by  astronomic  determi- 
nations. It  is  seen  that  if  the  topography  is  known  and  its  influence 
evaluated,  and  sufficient  observations  are  reduced,  the  distribution 
of  subcrustal  densities  and  consequently  the  amount  of  crustal 
strains  form  soluble  but  complex  problems. 

The  mathematical  mode  of  investigation  of  such  problems  has, 
however,  both  its  advantages  and  disadvantages.  The  advantages 
lie  in  giving  quantitative  results  and  in  the  test  of  the  accuracy  of 
the  trial  hypotheses  by  means  of  the  method  of  least  squares.  A 
disadvantage  lies  in  the  necessity  of  erecting  simple  hypotheses  in 
place  of  the  complex  realities  of  nature,  in  order  to  bring  the  data 
within  the  range  of  mathematical  treatment.  The  precision  of 
mathematical  analysis  is  furthermore  likely  to  obscure  the  lack  of 
precision  in  the  basal  assumptions  and  through  the  apparent 
finality  of  its  results  tends  to  hide  from  sight  other  possibilities  of 
the  solution. 

It  is  because  of  the  geologic  nature  of  the  hypotheses  on  which 
the  calculations  concerning  isostasy  rest,  and  the  geologic  bearing 
of  the  results,  that  it  is  no  act  of  presumption  for  the  geologist  to 
enter  into  this  particular  field  of  the  geodesist. 

The  measurements  of  isostasy  have  been  placed  most  fully  on 
a  quantitative  basis  by  Hayford,  and  the  science  of  geology  is  in- 
debted to  him  in  large  measure.  In  the  following  consideration  of 
the  geodetic  evidence  attention  will  be  confined  almost  entirely  to 
his  work,  supplemented  by  that  of  Bowie.  Hayford  was  the  first 
to  consider  the  influence  of  the  topography  and  its  compensation 


148  JOSEPH  BARRELL 

to  very  great  distances  from  each  station,  the  first  to  make  a  con- 
siderable number  of  trial  solutions  upon  various  assumptions  as  to 
the  depth  of  the  zone  of  isostatic  compensation,  with  the  result  that 
the  reduction  of  the  observations  gave  the  dimensions  of  the  earth 
with  a  considerably  smaller  probable  error  than  any  previous 
computations.1 

But  the  conclusions  in  regard  to  the  strength  of  the  crust,  drawn 
in  the  first  part  of  this  article  from  the  study  of  deltas,  stand  in 
strong  contrast  to  certain  statements  by  Hayford  and  later  by 
Hayford  and  Bowie.  This  second  part  must  therefore  outline  the 
results  reached  by  them  and  show  what  reconsiderations  are  neces- 
sary in  order  to  bring  into  harmony  their  conclusions  and  the 
evidence  derived  from  the  previous  geologic  study.  A  preliminary 
review  without  criticism  is  given  of  their  work  in  order  to  bring 
out  their  methods  and  results,  and  the  geologic  conclusions  which 
they  draw  from  those  results.  It  is  followed  by  a  re-examination  of 
the  subject  of  regional  versus  local  compensation.  This  is  the 
problem  of  the  size  of  the  area  over  which,  by  virtue  of  the  rigidity 
of  the  crust,  irregularities  of  density  and  topography  do  not  have 
individual  relationships  but  do  largely  compensate  each  other  over 
the  region  as  a  whole.  It  is  a  measure,  therefore,  of  the  areal  limits 
of  crustal  strength.  The  tests  employed  by  Hayford  and  Bowie 
are,  as  they  note,  indeterminate  up  to  radii  above  58.8  but  less 
than  166.7  km-  m  length.  Consequently  Hayford  did  not  change 
his  opinion,  based  upon  previous  investigations,  that  regional  com- 
pensation was  limited  to  areas  of  less  than  one  square  degree.  In 

1  The  final  publications  have  been  issued  by  the  United  States  Coast  and  Geodetic 
Survey  and  are  as  follows:  Hayford,  "The  Figure  of  the  Earth  and  Isostasy  from 
Measurements  in  the  United  States  (up  to  1906),"  1909;  referred  to  in  this  paper  as 
Hayford,  1906;  Hayford,  "Supplementary  Investigation  in  1909  of  the  Figure  of  the 
Earth  and  Isostasy,"  1910;  referred  to  in  this  paper  as  Hayford,  1909;  Hayford  and 
Bowie,  "The  Effect  of  Topography  and  Isostatic  Compensation  upon  the  Intensity  of 
Gravity,"  1912;  referred  to  in  this  paper  as  Hayford  and  Bowie,  1912;  Bowie,  "Effect 
of  Topography  and  Isostatic  Compensation  upon  the  Intensity  of  Gravity"  (second 
paper),  1912;  referred  to  in  this  paper  as  Bowie,  1912. 

In  addition  Bowie  has  published  in  the  American  Journal  of  Science,  "Some 
Relations  between  Gravity  Anomalies  and  the  Geologic  Formations  in  the  United 
States,"  (4)  XXXIII  (1912),  237-40. 

The  following  discussion  of  their  geodetic  measurements  and  results  will  be  con- 
fined to  the  work  in  these  five  papers. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  149 

this  article,  however,  two  other  tests  are  applied  which  indicate 
that  although  in  some  areas  compensation  does  not  extend  to  166 . 7 
km.  radius,  in  other  areas  it  extends  farther.  It  is  concluded  that 
the  United  States  shows  regional  departures  from  isostasy  over 
areas  many  times  larger  than  Hay  ford  thought  to  exist,  as  broad 
and  in  some  regions  probably  somewhat  broader  than  the  areas  of 
the  Nile  and  Niger  deltas,  the  breadth  depending  in  considerable 
part  upon  the  magnitude  of  the  loads  per  unit  of  surface.1 

GEODETIC    MEASUREMENTS    OF    ISOSTASY    BY    HAYFORD    AND    BOWIE 

Hayford's  conclusions  from  deflections  of  the  vertical. — The  posi- 
tions of  many  stations  over  the  United  States  were  determined  with 
great  accuracy  by  geodetic  measurements  from  other  stations,  thus 
making  a  closed  network.  The  positions  were  also  determined  by 
astronomic  observation.  The  differences  in  latitude  and  longitude 
between  the  geodetic  and  astronomic  positions  give  the  observed 
deflections  of  the  vertical  due  to  the  attraction  of  the  surface 
irregularities  and  internal  heterogeneities  of  the  geoid.  To  account 
for  these  deflections  the  gravitative  attraction  upon  the  plumb-line 
at  each  station  of  all  the  topography  from  ocean  bottoms  to  moun- 
tain tops  within  4,126  km.  was  computed.  The  influence  of  the 
topography  alone  upon  the  direction  of  the  vertical  is  known  as  the 
topographic  deflection  and  averages  a  little  over  30".  The  average 
of  the  actually  observed  deflections  are,  however,  but  a  fraction  of 
this  value.  Consequently  the  excesses  of  volume  represented  by 
continents  above  oceans,  and  by  plateaus  on  continents  must  be 
very  largely  balanced  and  neutralized  by  corresponding  deficiencies 
of  density  in  the  crust  beneath,  which  in  turn  explains  how  the 
larger  relief  is  sustained.  This  is  the  theorem  of  isostasy.  Various 
hypotheses  in  regard  to  the  magnitude  and  distribution  of  these 
deficiencies  in  density  under  the  continents,  of  excesses  under  the 
oceans,  may  be  made,  and  the  deflections  recomputed  on  these 
successive  suppositions  and  compared  with  the  observed  deflections. 

*At  the  recent  meeting  of  the  Geological  Society  of  America,  December  30,  1913, 
to  January  i,  1914,  Professor  W.  H.  Hobbs  gave  a  paper  on  "A  Criticism  of  the 
Hayfordian  Conception  of ' Isostasy  Regarded  from  the  Standpoint  of  Geology." 
The  writer  did  not  have  the  pleasure  of  hearing  this  paper,  but  it  is  clear  that 
Professor  Hobbs  has  attacked  independently  the  same  problems  as  here  discussed. 


150  JOSEPH  BARRELL 

The  difference  is  the  residual  error  due  to  the  partial  incorrectness 
of  a  hypothesis.  The  exactly  correct  hypothesis  would  reduce  all 
residual  errors  to  zero  except  for  the  errors  of  observation  and 
computation.  A  hypothesis  which  approximates  to  the  truth  will 
give  small  residual  errors.  In  a  large  mass  of  data  the  sum  of  the 
squares  of  the  residuals  as  derived  from  different  hypotheses  serves 
as  a  test  of  the  relative  agreement  of  the  hypotheses  with  nature, 
that  hypothesis  applying  best  for  which  the  sum  of  the  squares  is  a 
minimum.  In  all  of  the  complete  solutions  a  uniform  distribution 
of  compensation  was  assumed  to  exist  from  the  surface  to  the  bottom 
of  the  zone  of  isostatic  compensation.  That  is,  if  the  column  under 
a  certain  portion  of  land  was  3  per  cent  lighter  than  under  a  certain 
portion  of  water,  then  it  was  assumed  that  at  any  and  every  depth 
the  two  columns  differed  in  density  by  3  per  cent.  The  differences 
abruptly  terminate  at  the  level  where  the  two  columns,  the  long  but 
light  land  column  and  the  short  but  heavy  sea  column,  become  of 
equal  weight.  At  the  level  of  this  surface  isostatic  compensation  is 
complete  and  there  is  hydrostatic  equilibrium. 

A  tabulation  of  the  probabilities  of  these  hypotheses  as  applied 
to  the  whole  of  the  United  State  is  as  follows: 

TABLE  III 

Hypothesis  Sum  of  Squares  of  765  Residuals 

Solution  B  (extreme  rigidity;  depth  of  compensation  infinite) 107,385 

Solution  E  (depth  of  compensation  162. 2  km.) 105297 

Solution  H  (depth  of  compensation  120. 9  km.) 10,063 

Solution  G  (depth  of  compensation  113. 7  km.) 10,077 

Solution  A  (depth  of  compensation  zero) 18,889 

The  first  investigation,  that  of  1906,  favored  Solution  G,  the 
final,  that  of  1909,  as  shown  in  this  table,  favored  H.  The  most 
probable  depth  on  the  hypothesis  of  uniform  compensation  with 
depth  and  of  equal  depth  of  compensation  for  the  whole  United 
States  was  a  little  greater,  being  122.2  km.,  76  miles.  It  is  seen, 
however,  that  there  is  but  little  change  in  the  sum  of  the  squares  for 
a  considerable  range  in  the  assumed  depth.  Further,  Hayford 
states  that  the  hypothesis  of  all  compensation  being  attained  in  a 
lo-mile  stratum  whose  bottom  is  at  a  depth  of  35  miles  is  about  as 
probable  as  the  solution  which  he  adopted.1  Other  variations  in  the 
hypothesis  are  also  possible  with  about  the  same  probable  error.2 

1 1906,  p.  151.  a  1906,  p.  153. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST       151 

A  distribution  suggested  by  Chamberlin,  of  compensation  greatest 
a  little  below  the  surface  and  diminishing  to  nothing  at  178.6  miles, 
is  also  about  as  probable.  Hayford  therefore  does  not  claim  that 
his  geodetic  studies  determine  with  precision  the  nature  or  depth  of 
the  distribution  of  compensation.  The  figure  of  76  miles  should 
therefore  be  used  always  with  this  reservation. 

The  residuals  were  classified  into  fourteen  geographic  groups. 
The  most  probable  depths  of  compensation  indicated  for  the  several 
groups  range  from  66  to  305  km.  According  to  Hayford,  the 
evidence  from  these  groups  is,  however,  so  weak  and  conflicting  that 
he  sees  no  indication  that  the  depth  of  compensation  is  not  constant 
over  the  whole  area  investigated.1  He  notes  that,  so  far  as  the 
evidence  goes,  it  indicates  the  depth  of  compensation  to  be  greater 
in  the  eastern  and  central  portions  of  the  United  States  than  in  the 
western  portion.2  The  subject  is  one  which  will  be  taken  up  later 
in  the  discussion  of  geodetic  results. 

In  regard  to  the  completeness  of  compensation,  Hayford  states: 

From  the  evidence  it  is  safe  to  conclude  that  the  isostatic  compensation  is 
so  nearly  complete  on  an  average  that  the  deflections  of  the  vertical  are  thereby 
reduced  to  less  than  one-tenth  of  the  mean  values  which  they  would  have  if  no 
isostatic  compensation  existed.  One  may  properly  characterize  the  isostatic 
compensation  as  departing  on  an  average  less  than  one-tenth  from  completeness 
or  perfection.  The  average  elevation  of  the  United  States  above  mean  sea-level 
being  about  2,500  feet,  this  average  departure  of  less  than  one-tenth  part  from 
complete  compensation  corresponds  to  excesses  or  deficiencies  of  mass  repre- 
sented by  a  stratum  only  250  feet  (76  meters)  thick  on  an  average.3 

It  is  not  intended  to  assert  that  every  minute  topographic  feature,  such,  for 
example,  as  a  hill  covering  a  single  square  mile,  is  separately  compensated.  It 
is  believed  that  the  larger  topographic  features  are  compensated.  It  is  an 
interesting  and  important  problem  for  future  study  to  determine  the  maximum 
size,  in  the  horizontal  sense,  which  a  topographic  feature  may  have  and  still  not 
have  beneath  it  an  approximation  to  complete  isostatic  compensation.  It  is 
certain,  from  the  results  of  this  investigation,  that  the  continent  as  a  whole  is 
closely  compensated,  and  that  areas  as  large  as  states  are  also  compensated. 
It  is  the  writer's  belief  that  each  area  as  large  as  one  degree  square  is  generally 
largely  compensated.  The  writer  predicts  that  future  investigations  will  show 
that  the  maximum  horizontal  extent  which  a  topographic  feature  may  have  and 
still  escape  compensation  is  between  i  square  mile  and  i  square  degree.  This 
prediction  is  based,  in  part,  upon  a  consideration  of  the  mechanics  of  the 
problem.4 

1 1909,  pp.  55-59.  3 19o9,  p.  59. 

2 1906,  pp.  143, 146.  4 1906,  p.  169. 


152  JOSEPH  BARRELL 

These  conclusions  imply  a  weakness  of  the  crust  surprising  to  the 
geologist  and  stand  in  marked  contrast  to  those  figures  derived  from 
the  study  of  the  deltas  of  the  Nile  and  Niger.  This  subject  also  will 
be  discussed  later,  as  here  it  is  desired  to  give  only  a  summary  state- 
ment of  the  methods  and  conclusions. 

Hay  ford  and  Bowie  on  variations  of  gravity. — Regarding  the  rela- 
tions of  variations  in  gravity  to  isostasy,  Hayford  and  Bowie  state: 

As  soon  as  it  was  evident  that  the  proper  recognition  of  isostasy  in  connec- 
tion with  computations  of  the  figure  and  size  of  the  earth  from  observed 
deflections  of  the  vertical  would  produce  a  great  increase  in  accuracy,  it 
appeared  to  be  very  probable  that  a  similar  recognition  of  isostasy  in  connection 
with  computations  of  the  shape  of  the  earth  from  observations  of  the  intensity 
of  gravity  would  produce  a  similar  increase  of  accuracy.  Logically  the  next 
step  to  be  taken  was  therefore  to  introduce  such  a  definite  recognition  of  isostasy 
into  gravity  computations.  Moreover,  it  appeared  that  if  this  step  were  taken 
it  would  furnish  a  proof  of  the  existence  of  isostasy  independent  of  the  proof 
furnished  by  observed  deflections  of  the  vertical,  and  would  therefore  be  of 
great  value  in  supplementing  the  deflection  investigations  and  in  testing  the 
conclusions  drawn  from  them.  In  other  words,  the  effects  of  isostasy  upon  the 
direction  of  gravity  at  various  stations  on  the  earth's  surface  having  been 
studied,  it  then  appeared  to  be  almost  equally  important  to  investigate  the 
effects  of  isostasy  upon  the  intensity  of  gravity.1 

In  order  to  make  the  computations,  the  isostatic  compensation 
was  assumed  to  be  complete  under  every  topographic  feature  and 
uniformly  distributed  to  a  depth  of  114  km.  below  sea-level,  pro- 
ducing hydrostatic  equilibrium  at  this  depth.  The  mean  density  of 
2 . 67  was  taken  as  applying  to  the  whole  zone  to  this  depth.  Under 
land  3  km.  high  this  gives  a  density  of  2 . 60  from  sea-level  to  a  depth 
of  114  km.;  under  ocean  5  km.  deep  a  density  of  2.74  from  ocean 
bottom  to  114  km.  below  the  bottom.2 

The  authors  show  that  the  topography  and  its  compensation  for 
the  whole  earth  must  be  taken  into  consideration.  On  these 
assumptions  the  theoretic  value  of  gravity  was  computed  for  every 
station,  124  in  the  final  publication.  This  computed  value  is  sub- 
tracted from  the  observed  value  and  gives  the  " new-method" 
anomaly  for  each  station.  The  results  are  shown  in  Fig.  5. 

1  Hayford  and  Bowie,  1912,  p.  5. 

2  Hayford  and  Bowie,  1912,  pp.  9,  10. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST 


154  JOSEPH  BARRELL 

Of  the  two  other  principal  methods  of  gravity  reduction  which 
have  been  previously  used,  the  Bouguer  reduction  takes  no  account 
of  isostatic  compensation,  postulating  a  high  rigidity  of  the  earth's 
crust,  and  neglects  all  curvature  of  the  sea-level  surface.  The 
"free-air"  reduction  assumes  that  each  piece  of  topography  is 
compensated  for  at  zero  depth.  These  two  reductions  correspond 
thus  to  the  limiting  solutions  tried  for  deflections  of  the  vertical. 
The  sum  of  the  squares  of  the  new  method  anomalies,  when  com- 
pared respectively  with  the  similar  sums  derived  from  the  hypothesis 
of  rigidity  and  the  hypothesis  of  compensation  at  depth  zero,  shows 
that  the  assumption  of  isostatic  compensation  uniformly  dis- 
tributed to  a  depth  of  114  km.  gives  on  the  average  smaller 
anomalies;  is  therefore  much  more  probable  and  yields  a  more 
accurate  value  for  the  intensity  of  gravity.  The  mean  anomaly  of 
all  stations  in  the  United  States  without  regard  to  sign,  omitting  the 
exceptionally  large  anomalies  of  the  Seattle  stations,  is  as  follows: 

New  method o.  018  dyne1 

Bouguer o.  063 

Free  air o.  028 

The  value  of  gravity  for  the  United  States  Coast  and  Geodetic 
Survey  office  at  Washington  was  determined  as  980 .112  dynes  per 
gram.  The  mean  new-method  anomaly  is  consequently  about 
o .  00002  of  the  value  of  gravity.  The  probable  error  of  observation 
and  computation  is  about  0.003  dyne.  The  errors  may,  however, 
frequently  exceed  o .  004  dyne  and  in  rare  cases  may  be  as  great  as 
o.oio  dyne.2  The  fact  that  these  measures  of  gravity  are  the 
forces  acting  on  one  gram  will  be  understood  through  the  rest  of 
the  paper. 

Of  the  124  stations,  32  have  anomalies  between  o.  020  and  o. 030, 
12  have  anomalies  between  0.030  and  0.040.3  Still  smaller  num- 
bers of  stations  have  higher  anomalies.  These  anomalies  measure 
departures  in  the  earth's  crust  from  the  conditions  of  isostasy  which 
were  postulated.  In  the  interpretation  of  the  anomalies  in  terms 
of  mass  it  is  shown  that  a  small  excess  of  mass  immediately  below 

1  Bowie,  1912,  p.  12. 

'Hayford  and  Bowie,  1912,  p.  79;  Bowie,  1912,  p.  13. 

3  Bowie,  1912,  p.  13. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  155 

the  station  or  a  large  excess  at  great  depth  or  to  one  side  may  have 
the  same  effect.  Therefore  it  is  necessary  to  speak  of  the  net 
effective  excess  or  deficiency  of  mass.1  A  table  is  given  showing 
these  relations,  and  as  a  mean  working  hypothesis  it  is  assumed  that 
ordinarily  each  0.0030  dyne  of  anomaly  is  due  to  an  excess  or 
deficiency  of  mass  equivalent  to  a  stratum  100  ft.  thick.  In  the 
final  paper  it  is  concluded : 

From  the  evidence  given  by  deflections  of  the  vertical  the  conclusion  has 
been  drawn  that  in  the  United  States  the  average  departure  from  complete 
compensation  corresponds  to  excesses  or  deficiencies  of  mass  represented  by  a 
stratum  only  250  feet  thick  on  an  average.  The  gravity  determinations 
indicate  this  average  to  be  630  feet  instead  of  250  feet.  In  neither  case  is  the 
average  value  determined  or  defined  with  a  high  grade  of  accuracy.  The 
difference  between  the  two  determinations  of  the  average  value  is  therefore  of 
little  importance.  The  determination  given  by  the  gravity  observations  is 
probably  the  more  reliable  of  the  two.  Each  determination  is  significant 
mainly  as  showing  that  the  isostatic  compensation  is  nearly  perfect. 

The  average  elevation  in  the  United  States  above  mean  sea  level  is  about 
2,500  feet.  Therefore,  from  gravity  observations  alone  the  compensation  may 
be  considered  to  be  about  75  per  cent  complete  on  an  average  for  stations  in  the 
United  States.8 

This  conclusion  implies  a  somewhat  greater  rigidity  to  the  crust 
than  that  which  is  stated  for  the  deflections  of  the  vertical,  but  in 
regard  to  the  maximum  horizontal  extent  which  a  topographic 
feature  may  have  and  still  escape  compensation  the  authors  still 
express  the  belief  that  the  limit  is  between  one  square  mile  and  one 
square  degree.  "It  appears  from  the  inconclusive  evidence  fur- 
nished by  the  gravity  observations  that  the  radius  of  this  area  is 
probably  less  than  18.8  kilometers."3 

This  review  of  the  work  of  Hay  ford  on  deflections  of  the  vertical, 
and  of  Hayford  and  Bowie  on  the  gravity  anomalies  has  been  given 
in  order  that  the  methods  of  the  work,  its  bearings  on  the  strength 
of  the  crust,  and  the  conclusions  which  were  reached,  may  be  per- 
ceived. It  is  seen  that  a  large  difference  of  view  as  to  the  strength 
of  the  crust  exists  between  this  interpretation  from  the  geodetic 
evidence  and  that  from  the  geologic.  In  the  following  pages  will  be 

1  Hayford  and  Bowie,  1912,  pp.  108-12;  Bowie,  1912,  p.  22. 

2  Bowie,  1912,  pp.  22,  23.  3  Hayford  and  Bowie,  1912,  p.  102. 


156  JOSEPH  BARRELL 

given  a  discussion  which  it  is  thought  brings  out  certain  errors  in  the 
conclusions  drawn  from  the  geodetic  work  and  thereby  reconciles 
the  two  lines  of  evidence. 

REGIONAL  VERSUS   LOCAL  DISTRIBUTION   OF   COMPENSATION 

Conclusions  on  this  topic  by  Hay  ford  and  Bowie. — Under  this 
heading  Hayford  and  Bowie  state: 

The  question  whether  each  topographic  feature  is  completely  compensated 
for  by  a  defect  or  excess  of  mass  exactly  equal  in  amount  directly  under  it,  or 
whether  the  topographic  feature  is  compensated  for  by  a  defect  or  excess  of 
mass  distributed  through  a  more  extensive  portion  of  the  earth's  crust  than 
that  which  lies  directly  beneath  it,  is  a  very  important  one.  The  theory  of  local 
compensation  postulates  that  the  defect  or  excess  of  mass  under  any  topographic 
feature  is  uniformly  distributed  in  a  column  extending  from  the  topographic 
feature  to  a  depth  of  113.7  kilometers  below  sea  level.  The  theory  of  regional 
compensation  postulates,  on  the  other  hand,  that  the  individual  topographic 
features  are  not  compensated  for  locally,  but  that  compensation  does  exist  for 
regions  of  considerable  area  considered  as  a  whole. 

In  order  to  have  local  compensation  there  must  be  a  lower  effective  rigidity 
in  the  earth's  crust  than  under  the  theory  of  regional  compensation  only.  In 
the  latter  case  there  must  be  sufficient  rigidity  in  the  earth's  crust  to  support 
individual  features,  such  as  Pikes  Peak,  for  instance,  but  not  rigidity  enough  to 
support  the  topography  covering  large  areas. 

Certain  computations  have  been  made  to  ascertain  which  is  more  nearly 
correct,  the  assumption  of  local  compensation  or  the  assumption  of  regional 
compensation  only.  In  making  such  computations  it  is  necessary  to  adopt 
limits  for  the  areas  within  which  compensation  is  to  be  considered  complete. 
A  reconnoissance  showed  that  the  distant  topography  and  compensation  need 
not  be  considered,  for  their  effect  would  be  practically  the  same  for  both  kinds 
of  distribution.  As  a  result  of  this  reconnoissance  it  was  decided  to  make  the 
test  for  three  areas,  the  first  extending  from  the  station  to  the  outer  limit  of 
zone  K  (18. 8  kilometers),  the  second  from  the  station  to  the  outer  limit  of  zone 
M  (58.8  kilometers),  and  the  third,  to  the  outer  limit  of  zone  O  (166.7 
kilometers)  .* 

The  average  anomaly  with  regard  to  sign  by  the  new  method  with  local 
compensation,  and  the  average  anomaly  by  each  of  the  three  new-method 
reductions  with  regional  distribution  of  the  compensation  are  respectively 
— 0.002,  — o.ooi,  — o.ooi,  and  —0.002  dyne.  The  means  without  regard  to 
sign  for  the  different  distributions  of  the  compensation  are  respectively, 
0.020,  0.019,  0.019,  and  0.020  dyne.  These  mean  anomalies  give  only 
negative  evidence.2 

1  Hayford  and  Bowie,  1912,  p.  98.  2  Bowie.  1912,  p.  22. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  157 

The  problem  may  be  tested  in  another  way. 

If  local  compensation  be  true,  an  unusually  high  mountain  is 
underlain  by  unusually  light  matter  and  the  intensity  of  gravity  at 
a  station  on  its  top  is  less  than  if  the  mountain  was  supported  by 
regional  compensation  and  had  matter  of  the  mean  regional  density 
below  it. 

If  the  station  is  much  below  the  average  level  of  a  mountainous 
region,  local  compensation  implies,  on  the  contrary,  denser  matter 
beneath  and  a  higher  value  of  gravity  than  would  be  given  by 
regional  compensation.  These  relations  result  in  the  following 
principle :  For  stations  above  the  mean  level,  if  local  compensation 
be  nearer  the  truth  the  hypothesis  of  regional  compensation  would 
tend  to  show  its  error  by  large  negative  anomalies.  If  regional 
compensation  be  nearer  the  truth,  the  hypothesis  of  local  compensa- 
tion would  tend  to  show  its  error  by  giving  large  positive  anomalies. 
For  stations  below  the  mean  level  the  reverse  would  be  true.  But 
for  any  individual  station  other  departures  from  the  truth  of  that 
hypothesis  of  isostasy  which  gives  the  basis  for  the  calculations  may 
have  greater  influences  and  give  larger  anomalies  than  the  question 
to  be  tested.  Following  this  principle  it  is  stated: 

There  are  22  stations  in  the  United  States  in  mountainous  regions  and 
below  the  general  level  and  the  means,  with  regard  to  sign,  of  the  anomalies 
by  the  four  methods  of  distribution  are  o.ooo,  +0.001,  +0.003,  and  +0.005 
dyne,  while  the  means  without  regard  to  signs  are  respectively  0.017,  0.017, 
o  .018,  and  o .  019  dyne.  For  the  18  stations  in  the  United  States  in  mountain- 
ous regions  and  above  the  general  level  the  means,  with  regard  to  sign,  of  the 
anomalies  by  the  several  methods  of  distribution  of  the  compensation  are 
+0.003,  +0-003,  o.ooo,  and  — o.io  dyne.  The  means,  without  regard  to 
sign,  are  respectively  0.018,  0.018,  0.017,  and  0.020  dyne. 

The  mean,  with  regard  to  sign,  of  the  anomalies  for  the  stations  at  each 
of  the  two  mountain  groups,  indicates  that  the  theory  of  regional  distribution 
of  compensation  to  the  outer  limit  of  zone  O,  166.7  kilometers  is  far  from  the 
truth.  So  far  as  may  be  judged  from  the  other  average  anomalies  no  one 
method  seems  to  have  any  decided  advantage  (see  pp.  98-102  of  Special 
Publication  No.  lo).1 

Review  and  analysis  of  the  evidence. — -The  present  writer  does  not 
see  in  these  computations  any  support  for  the  hypothesis  of  local 

1  Bowie,  1912,  p.  22. 


158  JOSEPH  BARRELL 

compensation  of  the  topography  to  between  limits  of  one  square 
mile  and  one  square  degree  with  the  added  suggestion  of  a  radius  less 
than  1 8. 8  km.,  which  has  been  advanced  on  other  pages  by  the 
authors.1  These  figures  merely  show  that,  to  the  outer  limit  of 
zone  M,  radius  58.8  km.,  and  probably  to  outer  limit  of  zone  N, 
radius  99  km.,  one  method  is  as  good  as  another  for  purposes  of 
computation,  which  is  not  true  in  nature.  The  errors  introduced  by 
observation  and  computation,  the  errors  introduced  by  the  lack  of 
recognition  necessary  in  the  preliminary  hypothesis  regarding  the 
irregularities  in  the  depth  and  distribution  of  compensation — -these 
produce  effects  which  overshadow  the  small  systematic  differences 
due  to  the  hypotheses  of  local  versus  regional  compensation.  For 
the  outer  limit  of  zone  0,  radius  of  166.7  km.,  a  real  distinction 
does,  however,  begin  to  appear  in  the  data  for  the  two  groups  of 
mountain  stations.  It  is,  however,  very  small  and  based  upon  a 
rather  too  limited  number  of  stations  to  give  quantitative  reliability 
to  the  mean.  Furthermore,  as  discussed  in  detail  under  a  later 
heading,  there  is  quite  possibly  a  real  difference  between  the  limits 
of  regional  compensation  and  depth  of  compensation  in  the  moun- 
tain regions  of  the  West  compared  to  other  parts  of  the  continent. 
Evidence  drawn  from  the  Cordillera  cannot,  therefore,  be  applied 
safely  to  the  other  portions  of  the  United  States. 

Let  the  assumption  be  introduced  that  the  limits  of  regional 
compensation  are  variable,  ranging  from  100  to  500  km.  in  radius. 
Such  variable  limits  may  well  exist  because  of  several  factors;  first* 
because  of  a  real  variability  in  the  strength  of  the  crust;  second, 
because  the  greater  vertical  stresses  could  be  carried  only  by  smaller 
areas.  In  regions  of  mountainous  relief  due  to  folding,  or  of  high 
anomalies  due  to  great  irregularities  of  density,  the  mean  size  of 
unit  areas  should  therefore  be  less.  On  the  whole  the  anomalies  as 
well  as  the  relief  appear  to  be  somewhat  greater  over  the  western 
United  States.  Third,  in  regions  of  recent  block  faulting  or  warping 
the  stresses  have  presumably  been  lessened  from  what  they  were 
immediately  before  the  movement.  Such  diminution  of  strain 
could  take  place  by  the  breaking-up  of  a  large  unit  area  of  crust 
into  smaller  units  with  differential  movement  among  them,  as 

1  Hayford  and  Bowie,  1912,  p.  102. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  159 

well  as  by  vertical  movement  of  the  whole  area  to  a  level  best 
satisfying  the  stress.  The  western  United  States  is  known  to  be 
such  a  region,  which  in  the  late  Tertiary  and  up  to  the  present  has 
been  markedly  affected  by  block  faulting  and  differential  vertical 
movements. 

Suppose,  then,  that  the  mean  radius  of  regional  compensation  in 
a  mountainous  region  is  300  km.  but  that  unit  areas  exist  ranging  in 
radius  from  100  to  500  km.  Of  mountain  stations  located  at 
random,  a  fraction  of  the  total  number  would  be  situated  within  or 
near  areas  where  regional  compensation  did  not  extend  to  166 . 7  km. 
Let  the  stations  be  divided  into  one  group  consisting  of  those  below 
the  mean  regional  elevation  and  another  group  above  the  mean 
regional  elevation.  Let  the  anomalies  be  computed  successively 
according  to  hypotheses  of  regional  compensation  to  successive 
limits  and  the  mean  of  the  group  for  each  limit  be  taken.  This  is 
the  test  applied  by  Hayford  and  Bowie.  It  has  been  seen  that  for 
radii  of  18.8  and  58.8  km.  the  results  are  indeterminate.  For  a 
larger  radius  the  group  anomaly  might  be  expected  to  show  an 
increase  as  soon  as  the  assumed  radius  exceeded  the  actual  radii  of 
a  part  of  the  areas.  Consequently,  if  the  hypothesis  be  true  that 
the  areas  of  regional  compensation  are  variable  in  size,  the  mean 
anomalies  of  the  two  groups  of  22  and  18  stations,  found  with  regard 
to  sign  to  be  +0.005  and  —  o.oio  respectively  for  radius  of  166.7 
km.,  do  not  show  that  regional  compensation  on  the  whole  does  not 
exist  to  those  limits.  It  may  indicate  only  that  some  areas  are  less 
than  that  radius.  The  mean  radius  of  regional  compensation  may 
be  166.7  km.  or  possibly  even  larger.  Other  tests  must  therefore 
be  sought  which  will  give  a  more  conclusive  answer. 

Further,  it  is  to  be  noted  that  the  mean  anomalies  with  regard 
to  sign  for  the  hypothesis  of  regional  compensation  to  radius  of 
166.7  km.,  although  somewhat  greater  than  for  the  other  hypoth- 
eses, are  yet  of  the  same  order  of  magnitude;  and  in  all  cases  are 
but  a  fraction  of  the  mean  anomaly  without  regard  to  sign.  Appar- 
ently, then,  the  assumption  of  regional  compensation  to  166.7  km. 
introduces  a  smaller  error  than  the  assumption  of  uniform  and  com- 
plete compensation  with  an  average  specific  gravity  of  2.67  to  a 
constant  depth  of  114  km. 


160  JOSEPH  BARRELL 

The  test  by  adjacent  stations  at  different  elevations. — -There  is, 
however,  another  way  of  using  the  data  given  for  stations  situated 
well  above  and  below  the  mean  elevation  of  mountainous  regions. 
If  a  pair  of  stations  be  taken  close  together,  one  far  above  the  mean 
elevation,  the  other  far  below,  they  will  presumably,  because  of 
their  juxtaposition,  be  affected  in  much  the  same  way  by  the  errors 
incident  to  the  hypothesis  of  uniform  compensation  through  a  depth 
of  114  km.,  with  complete  compensation  at  that  depth.  In  order 
that  good  results  may  be  obtained,  however,  the  specific  gravity  of 
the  local  rocks  should  be  carefully  determined  in  order  to  have  a 
correction  for  the  mass  between  the  stations.  The  parts  of  the 
anomalies  due  to  the  irregularities  and  incompleteness  of  compensa- 
tion will  ordinarily  have  the  same  sign  and  be  of  nearly  the  same 
value  at  the  adjacent  stations.  This  is  indicated  by  the  contour 
lines  of  Fig.  5,  which  show 'that  in  the  same  region  the  anomalies 
are  of  sufficiently  regular  gradation  in  magnitude  to  make  the 
drawing  of  contour  lines  possible.  The  parts  of  the  anomalies  at 
the  high  and  low  stations  due  to  errors  in  the  hypothesis  of  local  or 
regional  compensation  will,  however,  be  of  opposite  sign.  If,  then, 
the  algebraic  difference  of  the  anomalies  for  such  a  pair  of  stations 
be  computed  for  successive  hypotheses  of  broader  regional  com- 
pensation, the  part  of  the  anomalies  due  to  vertical  imperfection  of 
the  hypothesis  will  be  largely  eliminated.  The  algebraic  difference 
measures  the  horizontal  imperfection  of  the  hypothesis.  That 
hypothesis  is  favored  whose  assumed  radius  of  regional  compensa- 
tion gives  a  minimum  value  to  this  algebraic  difference.  This  test 
may  be  made  by  combining  data  given  on  p.  ico,  Hayford  and 
Bowie,  with  p.  15,  Bowie;  although,  because  of  incompleteness  of 
the  tables,  this  combination  gives  the  data  for  only  a  few  of  the 
properly  situated  mountain  stations.  The  best  couple  of  stations 
for  the  application  of  this  test  consist  of  42,  Colorado  Springs,  and 
43  Pikes  Peak.  Somewhat  more  distant  stations — -44,  Denver,  and 
45,  Gunnison — may  also  be  added  to  the  group.  The  tabulation  is 
shown  on  p.  161  (Table  IV). 

It  is  seen  that  for  three  of  the  four  Colorado  stations  the  absolute 
value  of  the  anomaly  is  least  with  regional  compensation  to  166.7 
km.  For  the  fourth  station  it  remains  practically  constant  for 


THE  STRENGTH  OF  THE  EARTH'S  CRUST 


161 


all  the  cases.  The  anomalies  were  not  computed  for  greater  radii. 
The  more  convincing  argument,  however,  for  regional  compensation 
to  at  least  166 . 7  km.  radius  in  the  vicinity  of  Pikes  Peak  is  the  fact 
that  the  algebraic  difference  of  the  anomalies  between  the  top  and 
bottom  of  the  mountain,  stations  43  and  42,  is  less  than  one-half  for 
regional  compensation  to  166. 7  km.  radius  than  for  the  correspond- 
ing value  given  by  the  hypothesis  of  local  compensation.  The 
decrease  in  the  difference  is  furthermore  progressive  with  each 

TABLE  IV 


NUMBER  AND  NAME 
OF  STATION 

ELEVATION 
OF  STATION 
IN  METERS 

DISTANCE 
FROM  MEAN 
ELEVATION 
IN  METERS 

WITHIN 

100  MILES 

ANOMALY  WITH  REGIONAL  COMPENSATION 
WITHIN  OUTER  LIMIT  OF 

Local  Com- 
pensation. 
Radius 
o.o  Km. 

Zone  K, 
Radius 
18.8  Km. 

ZoneM, 
Radius 
58.8  Km. 

Zone  O, 
Radius 
166.7  Km. 

COLORADO 
42  .   Colorado 
Springs 

1,841 

4,293 
1,638 

2,340 

—  420 
+  2,035 
-574 
-380 

-458 

—  0.009 
+    -019 
—    .018 
+    .018 

-    .009 

—  0.009 
+    .Oil 
—    .Ol6 
+    .O2I 

—    .004 

—  o.oio 
+    .006 
-    .009 
+    .026 

+    .007 

—  O.OIO 

+    .002 
—    .001 
+    .016 

+    .005 

43.  Pikes  Peak  
44.  Denver  
45.  Gunnison  
Mean  of  42,  44,  and 

4C    . 

Algebraic  Difference 
43—42  

+    .028 
+    .028 

+    .020 
+    .015 

+    .016 
—    .001 

+     .012 

-    -003 

43—  (mean  of  42,  44, 
45)  

ARIZONA 
68.  Yavapai  
69.  Grand  Canyon.  . 

2,179 
849 

+  512 
—  824 

—     .001 
—     .OI2 

—     .001 

—   .on 

—     .001 
—     .Oil 

-    .009 

—     .021 

Algebraic   difference 
68-60 

+O.OII 

+O.OIO 

+O.OIO 

+0.012 

assumed  widening  of  the  zone.  The  result  of  adding  the  more 
distant  stations,  44  and  45,  favors  regional  compensation  more 
markedly  but  is  indeterminate  between  M  and  O.  It  would  seem, 
then,  that  the  front  range  of  the  Rocky  Mountains  in  Colorado  is 
upheld  above  the  surrounding  plains  and  parks  by  virtue  of  the 
rigidity  of  the  earth. 

The  two  stations  in  Arizona  at  68  and  69  are  well  situated  also 
to  test  the  question  of  local  versus  regional  compensation,  but  the 


162  JOSEPH  BARRELL 

difference  in  the  anomalies  in  this  case  is  so  nearly  constant  as  to 
give  an  indeterminate  answer.  In  the  absence  of  more  detailed 
statements  by  Hay  ford  and  Bowie  the  reason  why  the  anomaly  at 
the  Grand  Canyon  station  69  reaches  a  larger  negative  value  for 
regional  compensation  to  166.7  km.  than  for  more  limited  com- 
pensation is  not  evident.  The  usual  rule  is  that  the  progressive 
change  in  the  anomaly  for  stations  below  the  regional  level  for 
successive  assumptions  of  wider  regional  compensation  is  by 
increments  with  a  plus  sign.  Here,  on  the  contrary,  the  change  in 
the  limits  from  zone  M  to  zone  0  involves  a  minus  increment  of 
o.oio  in  the  anomaly.  The  cause  of  this  reversal  of  sign,  which 
the  writer  does  not  understand,  seems  in  this  case  to  be  the  cause  of 
the  indeterminate  result. 

Another  line  of  evidence  as  to  the  effective  limits  over  which 
the  rigidity  of  the  earth  may  extend  is  derived  from  a  study  of  the 
grouping  of  the  deflections  of  the  vertical  shown  in  illustrations 
2,  3,  5,  6,  Hayford,  1909,  and  the  lines  of  equal  anomaly  for  the  new 
method  of  reduction,  illustration  No.  2,  Bowie,  1912,  the  latter 
giving  the  basis  for  Fig.  5  of  this  article. 

The  test  by  areas  of  grouped  residuals. — Illustration  No.  5, 
Hayford,  1909,  shows  the  grouping  of  the  residuals  of  solution  H  for 
the  north  and  south  components  of  the  deflections.  An  area  with 
a  plus  sign  corresponds  to  an  excess  of  density  to  the  south,  or 
deficiency  to  the  north.  An  area  with  a  minus  sign  corresponds  to 
a  deficiency  of  density  to  the  south,  or  excess  to  the  north.  A 
north-south  chain  of  stations  is  therefore  best  for  ascertaining  the 
limits  of  the  areas  of  north-south  deflection  of  like  sign.  Such  a 
belt  extends  across  the  United  States  between  long.  97°  and  98°, 
showing  9  areas  covering  1,620  miles.  The  mean  intercept  is 
therefore  180  miles.  This  mean  intercept  must  be  somewhat  less 
than  the  mean  diameter. 

Illustration  No.  6,  Hayford,  1909,  shows  the  grouping  of  the 
residuals  of  solution  H  for  the  east  and  west  components  of  the 
deflections.  An  area  with  a  plus  sign  corresponds  to  an  excess  of 
density  to  the  east,  or  deficiency  to  the  west.  An  area  with  a 
minus  sign  corresponds  to  a  deficiency  of  density  to  the  east,  or 
excess  to  the  west.  An  east-west  chain  of  stations  is  therefore  best 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  163 

for  ascertaining  the  limits  of  the  areas  of  like  sign.  Such  a  belt 
extends  across  the  United  States  between  lat.  38°  and  39°. 

The  following  adjustments  in  groups  seem,  however,  fair  to 
make,  considering  the  lack  of  exact  accuracy  in  any  one  station. 
At  Cincinnati  is  a  station  showing  small  residuals  opposite  in  sign 
to  the  stations  on  each  side.  If  this  is  overlooked,  three  small 
groups  become  one  of  average  size.  In  central  Kansas  a  small 
minus  area  depending  on  a  single  observation  may  be  likewise 
omitted.  In  western  Colorado  several  small  areas  depending  each 
upon  two  observations  had  their  number  diminished  by  one.  The 
same  was  done  in  California.  This  gave  14  areas  extending  over 
2,580  miles,  a  mean  individual  intercept  of  184  miles.  If  16  areas 
be  taken,  a  mean  value  is  derived  of  161  miles.  More  weight,  it  is 
thought,  is  to  be  attached  to  the  determination  of  184  miles,  and 
this  is  supported  by  the  180  miles  shown  by  the  north-south  chain 
of  stations. 

The  areas  of  like  sign  are  between  centers  of  excess  and  defect  of 
mass.  They  are  not,  therefore,  coincident  with  the  areas  of  excess 
and  defect,  but  in  discussing  the  average  size  of  areas,  the  one  may 
be  used  as  a  measure  of  the  other. 

It  may  be  concluded,  therefore,  that  the  deflections  of  the  verti- 
cal show  areas  with  departures  from  isostatic  equilibrium  in  one 
direction  and  these  areas  average  about  180  miles,  290  km.,  in  mean 
intercept.  The  mean  diameters  of  the  areas  of  like  sign  are  pre- 
sumably somewhat  greater.  This  would  make  the  mean  radius  of 
areas  of  regional  compensation,  as  indicated  by  similarity  of  sign 
among  residuals,  at  least  166.7  ^m- — the  radius  of  the  outer  limit 
of  zone  O  used  in  the  discussion  of  the  gravity  anomalies. 

If  we  turn  now  to  the  anomalies  shown  by  the  determinations  of 
gravity,  Fig.  5,  adapted  from  Bowie,  shows  their  segregation  into 
areas  of  like  sign.  The  mean  value  without  regard  to  sign  for  all 
stations  excluding  Seattle  is  0.018  dyne  per  gram.  Including  the 
two  Seattle  stations  the  mean  is  0.020  dyne.  Between  the  con- 
tours for  —  0.020  and  +0.020  lie  tracts  where  the  anomalies  are 
within  the  mean  limits.  The  areas  of  exceptionally  large  anomalies 
are  above  those  limits.  It  is  only  these  which  form  on  this  illus- 
tration well-defined  inclosed  areas,  but  even  these  are  far  from 


164  JOSEPH  BARRELL 

regular  in  outline.  The  areas  showing  positive  anomalies  of  more 
than  0.020  dyne  were  estimated  roughly  to  average  130  by  240 
miles  across,  a  mean  diameter  of  175  miles.  The  areas  showing 
negative  anomalies  of  more  than  0.020  dyne  were  found  to  average 
roughly  about  190  by  250  miles,  a  mean  diameter  of  220  miles.  The 
long  narrow  connections  were  neglected  in  making  this  estimate. 
Unit  areas  of  more  than  mean  anomaly  may  therefore  be  taken  to 
average  about  200  miles  or  320  km.  in  diameter.  The  mean  radius 
is  therefore  approximately  that  of  the  outer  limits  of  zone  O, 
166.7  km- 

The  figures,  although  they  correspond  fairly  closely  to  those 
derived  from  the  deflections  of  the  vertical,  cannot  in  reality  be  very 
well  compared,  since  these  are  areas  selected  because  the  anomaly 
rises  above  a  certain  magnitude;  the  others  represent,  on  the  con- 
trary, a  succession  of  contiguous  areas  between  centers  of  excess 
and  defect  in  mass  without  reference  to  magnitude.  Apparently 
some  influence  blurs  out  the  limitations  of  areas  of.  small  gravity 
anomaly.  This  will  be  discussed  in  a  later  part. 

Now  assume  for  the  moment  that  isostatic  compensation  is  uni- 
form to  the  bottom  of  the  zone,  as  postulated  by  the  hypothesis; 
that  is,  that  the  residuals  and  anomalies  are  due  to  excesses  or 
defects  of  mass  which  are  uniformly  distributed.  Then,  over  any 
one  area  of  excess  or  deficiency  of  mass,  the  deflections  around  it 
and  anomalies  within  it  signify  a  departure  from  compensation  in 
one  direction.  This  is  a  regional  departure.  If  the  strength  of  the 
crust  was  so  small  that  it  was  able  to  support  notable  departures 
from  compensation  over  areas  of  only  one  square  degree  or  less, 
then  these  large  unit  areas  could  not  exist.  A  vertical  warping  up 
or  down  would  immediately  take  place  until  the  broad  region  as  a 
whole  lay  so  close  to  complete  compensation  that  its  surface 
irregularities  became  subdivided  into  subordinate  positive  and 
negative  areas  of  the  limiting  size.  The  sum  of  the  excesses  and 
defects  of  mass  would  approach  zero  in  broad  areas  containing 
many  unit  departures.  It  would  seem,  therefore,  that  the  geodetic 
results  shown  in  Fig.  5,  instead  of  indicating  local  compensation  to 
limits  of  less  than  one  square  degree,  show  on  the  contrary  a  ready 


THE  STRENGTH  OF  THE  EARTH'S  CRUST       165 

capacity  of  the  crust  under  the  United  States  to  carry  over  areas  of 
from  5  to  10  or  15  square  degrees,  and  exceptionally  over  even  larger 
areas,  departures  from  equilibrium  greater  than  the  mean.  This 
agrees  in  order  of  areal  magnitude  with  the  Nile  and  Niger  deltas. 
However,  the  influence  of  irregularity  in  the  distribution  of  com- 
pensation with  depth,  and  the  magnitude  of  stress  per  unit  area 
remain  to  be  investigated. 

[To  be  continued] 


THE  STRENGTH  OF  THE  EARTH'S  CRUST 


JOSEPH  BARRELL 
New  Haven,  Connecticut 

PART  III.    INFLUENCE  OF  VARIABLE  RATE  OF  ISOSTATIC 
COMPENSATION 

INTRODUCTION  AND  SUMMARY 209 

THE  SPECIFIC  GRAVITY  OF  ROCKS 211 

INTERPRETATION  OF  ANOMALIES  IN  TERMS  OF  MASS  AND  DEPTH  .  216 

RELATIONS  OF  ANOMALIES  TO  EXPOSED  GEOLOGIC  FORMATIONS  .  221 
LARGE  OUTSTANDING  ANOMALIES  NOT  RELATED  TO  GEOLOGY  OR 

TOPOGRAPHY 227 

CRITERIA  FOR  SEPARATING  VERTICALLY  IRREGULAR  COMPENSATION 

FROM  REGIONALLY  INCOMPLETE  COMPENSATION  .  .  .  .  228 
GRAVITY  ANOMALIES  CAUSED  LARGELY  BY  REGIONAL  DEPARTURES 

FROM  ISOSTASY 234 

INTRODUCTION  AND  SUMMARY 

The  work  of  Hayford  on  the  deflections  of  the  vertical,  and  of 
Hayford  and  Bowie  on  the  anomalies  of  gravity,  has  supplied  the 
geodetic  data  from  which  future  work  must  start.  As  an  initial 
basis  to  guide  their  work,  it  was  desirable  to  assume  the  hypothesis 
that  isostatic  compensation  was  complete  for  each  topographic 
irregularity,  giving  local  compensation,  and  that  it  was  uniformly 
distributed  to  a  constant  depth.  The  actual  results  may  then  be 
compared  to  this  ideal  of  local,  uniform,  and  complete  isostasy  and 
the  degree  of  departures  noted,  as  given  by  residuals  and  anomalies. 

In  Part  II  the  subject  of  the  regional  distribution  of  compensa- 
tion was  examined  and  the  conclusion  was  reached  that  the  crust 
was  sufficiently  rigid  to  bear  such  mountains  as  Pikes  Peak  without 
requiring  special  compensation  below.  In  general  it  is  thought 
compensation  in  mountain  regions  extends  to  more  than  200  km. 
and  in  some  regions  to  more  than  400  km.  In  this  part  are 
considered  the  effects  of  variations  in  the  vertical  distribution  of 

209 


210  JOSEPH  BARRELL 

compensation  and  the  degree  to  which  such  variability  may  give 
rise  to  anomalies  and  residuals  without  signifying  incompleteness 
of  compensation  in  the  column  as  a  whole  or  regional  departures 
from  isostasy. 

In  order  to  show  the  limits  of  variation  in  density  which  are  to 
be  expected,  the  specific  gravity  of  rocks  is  first  considered.  Figures 
are  computed  for  the  mean  specific  gravity  of  igneous  rocks  and 
the  three  types  of  sediments.  It  is  shown  that  the  range  of  varia- 
tion is  an  important  factor.  Under  the  subject  of  th«-  relations 
between  mass  and  the  distance  of  mass  upon  anomahV  ,  the  effects 
are  computed  of  unit  masses  at  various  depths  ai.u  extorting 
various  distances.1  This  lays  the  basis  for  considering  the  Influence 
of  the  specific  gravity  of  the  surface  geologic  formations  upon  thp> 
difference  between  the  mean  anomalies  for  stations  on  pre-Cambrian 
and  those  on  Cenozoic  areas.  It  is  found  that  the  greater  density 
of  the  older  rocks  accounts  for  a  part  and  another  part  is  accounted 
for  by  their  resistance  to  erosion.  This  still  leaves,  however,  large 
outstanding  regional  variations  not  related  to  surface  geology  or 
topography  and  requiring  some  other  explanation.  To  that  end 
criteria  are  discussed  for  the  recognition  and  separation  of  the 
effects  of  mere  variable  vertical  distribution  of  compensation  on 
the  one  hand,  from  partial  regional  absence  of  isostasy  on  the  other. 
It  is  concluded  from  the  application  of  these  criteria  that  the 
anomalies  are  in  large  part  caused  by  real  regional  departures 
from  isostasy  extending  over  broad  areas.  The  results  are  thus 

1  A  paper  by  Gilbert  has  recently  appeared  entitled  "Interpretation  of  Anomalies 
of  Gravity"  (Part  C,  Professional  Paper  85,  U.S.  Geological  Survey,  1913).  This 
did  not  reach  the  present  writer  until  after  Parts  III  and  IV  of  this  article  were  in 
galley  proof,  so  that  his  results  cannot  be  as  fully  interwoven  into  the  discussion  as 
would  otherwise  have  been  the  case.  On  pp.  30,  31,  Gilbert  considers  the  interpre- 
tation of  anomalies  on  the  assumption  of  vertical  heterogeneity  of  the  crust  and  shows 
clearly  that  moderate  variations  of  density  in  a  vertical  direction  could  explain  them. 
From  this  he  infers  that  the  anomalies  may  be  due  in  part  to  such  irregularities.  This 
is  the  topic  which  is  treated  in  Part  III  of  the  present  article  under  the  title  "Inter- 
pretation of  Anomalies  in  Terms  of  Mass  and  Depth."  The  method  of  reasoning  is 
somewhat  different,  but  although  the  conclusion  reached  is  the  same,  the  calculations 
given  here  are  intended  to  bring  out  in  addition  the  limitations  of  area  and  mass  within 
which  that  principle  applies.  It  is  concluded  as  a  result  of  the  following  examination 
of  the  evidence  that  although  vertical  variations  of  density  are  a  real  cause  they  are 
not  the  major  cause  of  anomalies. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  211 

confirmatory  of  those  reached  in  Parts  I  and  II.  In  addition,  how- 
ever, it  appears  that  there  is  a  regional  departure  from  isostasy  of 
two  orders  of  magnitude.  Loads  under  the  mean  value,  giving 
anomalies  below  0.018  to  0.020  dyne  and  estimated  to  be  equiva- 
lent to  about  750  feet  of  rock,  can  be  carried  over  regions  of  irregu- 
lar boundaries  ranging  up  to  from  1,000  to  2,000  km.  across.  Over 
such  a  broad  region  the  anomalies  are  of  one  sign  except  for  some 
smaller  well-defined  sub-areas  of  high  anomaly  within  them  which 
may  or  may  not  have  the  same  sign.  These  smaller  areas  give  a 
higher  order  of  stress  magnitude  and  are  of  more  restricted  dimen- 
sions, being  measured  in  hundreds  of  kilometers.  They  range  in 
magnitude  of  anomaly  to  several  times  the  value  of  the  mean  and 
the  equivalent  radii  of  their  areas  probably  average  100  to  200  km. 
The  deflection  residuals  show  by  the  limits  of  the  areas  of  like 
sign  that  the  regional  variations  of  gravity  anomalies  of  this 
areal  magnitude  extend  over  the  whole  country,  but  where  the 
amounts  of  the  local  anomalies  are  less  in  value  than  the  mean 
they  are  largely  masked  on  the  contour  map  of  gravity  anomalies 
(Fig.  5),  because  of  their  superposition  upon  the  broader  areas. 
Presumably  a  multiplication  of  the  gravity  stations  would  bring 
them  to  light  as  undulations  in  the  contours  which  show  the 
regional  departures. 

A  final  conclusion  on  the  subject  of  the  variable  vertical  distri- 
bution of  mass  must,  however,  be  deferred  until  consideration  has 
been  given  to  a  hypothesis  advanced  by  Gilbert  in  his  recent  paper, 
that  heterogeneities  of  mass  below  the  zone  of  compensation  may 
be  the  cause  in  major  or  minor  part  of  the  apparent  departures  from 
isostasy.  This  is  a  subject  too  large  to  be  considered  in  this  third 
part  of  the  present  article,  but  it  is  planned  to  investigate  it  in 
Part  V  by  a  method  of  graphic  analysis  devised  for  determining 
the  depth  of  excesses  or  deficiencies  of  mass. 

THE    SPECIFIC   GRAVITY   OF   ROCKS 

For  a  knowledge  of  the  variations  of  density  likely  to  occur  in 
rocks  it  is  important  to  know  the  range  in  specific  gravities  shown 
by  the  common  rock  types.  The  following  figures,  except  those  for 
shale,  are  taken  from  Pirsson's  Rocks  and  Rock  Minerals: 


212  JOSEPH  BARRELL 

TABLE  V 

Rock  Specific  Gravity 

Granite 2 . 63-2 . 75 

Syenite 2.6  -2.8 

Diorite 2.8  -3.1 

Dolerite 3.0-3.3 

Limestone 2.6-2.8 

Sandstone 2.5-2.7 

Shale 2.4  -2.8 

Slate About  2 . 8 

[The  specific  gravity  of  shale,  although  the  most  abundant  of  sedimentary 
rocks,  is  not  given  in  any  of  the  manuals  of  geology,  but  Professor  Hobbs,  who 
has  read  much  of  this  manuscript  and  to  whom  the  writer  is  indebted  for  a 
number  of  suggestions,  has  called  attention  to  the  above  figure  as  given  by 
Trautwine.  In  general,  Trautwine  and  Kent  give  a  somewhat  greater  range 
in  specific  gravities  and  they  average  a  little  lower  than  those  here  given.  The 
figures  from  Pirsson,  however,  probably  express  more  closely  the  relation  of  the 
petrologic  type  and  the  more  compact  states  of  rocks  to  their  density.  They 
are,  therefore,  thought  to  be  better  representative  of  the  lithosphere.] 

These  figures  show  that  notable  departures  may  occur  from 
the  mean  density  of  the  outer  crust  and  suggest  furthermore  that 
2.67,  the  mean  density  used  by  Hayford,  is  lower  than  the  actual 
mean.  A  more  thorough  analysis  of  the  subject  is  therefore  needed. 

The  abyssal  igneous  rocks  and  metamorphic  rocks  are  almost 
without  pore  space.  The  sedimentary  rocks,  on  the  other  hand, 
possess  abundant  pore  space  in  their  unconsolidated  states,  very 
little  in  their  compact  states.  The  latter  is  the  usual  mode  of 
occurrence  in  the  older  geological  formations.  The  density  is 
therefore  a  function  of  both  mineral  composition  and  porosity. 
The  chemical  compositions  of  the  several  rock  types  and  also  of 
the  average  sediment  and  the  average  igneous  rock  are  well  known. 
The  mineral  compositions  are  less  well  known  but  may  be  computed 
with  a  fair  degree  of  accuracy;  the  densities,  on  the  contrary,  are 
least  commonly  reported  and  the  mean  densities  of  the  rock  types 
cannot  in  consequence  be  closely  determined  by  averaging  numerous 
determinations,  as  is  done  for  the  chemical  compositions.  It  seems 
desirable,  therefore,  to  compute  the  densities  of  the  rock  types 
from  the  chemical  and  mineral  compositions,  combining  this  with 
the  densities  of  the  individual  minerals,  making  a  separate  correc- 


THE  STRENGTH  OF  THE  EARTH'S  CRUST 


213 


tion  for  the  porosity  factor.     The  data,  assembled  from  various 
sources1  and  subjected  to  computation,  give  the  following  results: 

TABLE  VI 

COMPOSITION  OF  AVERAGE  IGNEOUS  ROCK 

Mineral  Percentage 

Quartz . 12.0 

Feldspars 

Orthoclase  molecule 22.0 

Albite  molecule 29.5 

Anorthite  molecule 8.0 

Hornblende  and  pyroxene 16. 8 

Mica 3.8 

Accessory  minerals 7.9 


100. o 


TABLE  VII 

COMPOSITION  OF  AVERAGE  SEDIMENTS 


Mineral 

Shale 

Sandstone 

Limestone 

Quartz  

22.3* 

66.8* 

2.O 

Feldspars 
Orthoclase 

18  o 

7  ° 

o  •? 

Labradorite  
Clay 

12.  0 

2<    of 

4-5 
6  6f 

O.  I 
2    Ol 

Limonite        

S  6 

i  8 

o  6 

Calcite    1  

{cc  .0 

Dolomite/  

5-7 

ii.  i 

33 

35  -O 

Other  minerals  

ii  .4 

2.  2 

c.o 

IOO.O 

IOO.O 

IOO.O 

*  The  total  percentage  of  free  silica. 

t  Probably  sericite  in  part;  in  that  case  the  feldspar  figure  becomes  lower. 

J  Two  per  cent  clay  takes  o .  79  of  AUOj.  Thb  requires  that  most  of  alkalies  form  non-aluminous 
hydrous  silicates  or  that  0.81  AhOj  as  given  by  Clarke  is  too  low. 

It  is  thought  that  the  densities  without  porosity  are  figures  of 
some  value  for  geodetic  computations.  The  chief  error  in  making 
the  final  estimates  is  in  connection  with  the  lack  of  accurate  knowl- 
edge regarding  the  pore  space  of  those  sedimentary  rocks  not  used 

1  For  data  on  the  mean  chemical  and  mineral  composition  of  rocks  see  F.  W. 
Clarke,  "Data  of  Geochemistry,"  Bull.  491,  U.S.  Geol.  Surv.,  1911,  pp.  30,  31.  For 
specific  gravities  of  minerals  see  Pirsson,  Rocks  and  Rock  Minerals,  1908,  p.  31;  also 
Dana,  Mineralogy.  For  a  discussion  of  pore  space  see  Fuller,  "Total  Amount  of 
Free  Water  in  the  Earth's  Crust,"  Water  Supply  Paper  No.  160,  U.S.  Geol.  Surv., 
1906,  pp.  59-72. 


214  JOSEPH  BARRELL 

as  building  stones,  but  this  affects  appreciably  the  density  of  only 
a  superficial  layer  and  chiefly  of  the  youngest  deposits. 

The  ratio  of  shale,  sandstone,  and  limestone  in  the  average 
sediment  in  percentage  is,  according  to  Mead,1  shale  80,  sandstone 
n,  limestone  9.  The  ratio  of  average  porosities  in  percentage  is, 
according  to  Fuller,2  crystalline  rocks  0.2,  shales  4,  sandstones  15, 
limestones  5.  The  figure  given  by  Fuller  for  shale  rests  upon  a 
single  determination  of  7.8  per  cent  by  Delesse,  and  is  averaged 
in  by  Fuller  with  slate.  Eight  per  cent  porosity  will  here  be 
assumed  as  probably  a  better  estimate.  This  gives  the  porosity 
of  the  average  sedimentary  rock  as  8 . 5  per  cent.  The  pore  space 
may  be  taken,  following  Fuller's  estimate,  as  half  filled  with  water. 

From  these  data  the  specific  gravities  are  computed  to  be  as 
follows: 

TABLE  VIII 
SPECIFIC  GRAVITIES  COMPUTED  FROM  MINERAL  COMPOSITIONS 


Rock 

No  Pore  Space 
Allowed 

Pore  Space  Half 
Filled  with  Water 

Average  igneous  rock.  . 
Shale  
Sandstone  

2.80 
2.69 
2.67 

2.80* 

2-51 
2.35 

Limestone  
Average     sedimentary 
rock  

2.76 
2.70 

2.64 
2.50 

*The  same  figure  as  used  by  Chamberlin  and  Salisbury,  Geology,  I  (1904),  538;  also  by  Pirsson, 
Rocks  and  Rock  Minerals;  also  by  G.  H.  Darwin  as  the  density  of  the  outer  crust. 

Where  Cenozoic  deposits  occur  in  thickness,  they  are  consider- 
ably compacted  except  at  the  surface,  but  still  the  mean  specific 
gravity,  owing  to  the  abnormal  pore  space  and  deficiency  in  lime- 
stones, is  doubtless  less  than  2.50;  2.45  may  be  taken.  It  is 
probable,  on  the  other  hand,  that  the  Paleozoic  rocks  on  the  whole 
have  somewhat  less  pore  space  than  this  average,  especially  as  the 
porosity  figure  for  sandstone  rests  mainly  upon  determinations  for 
browns  tone,  a  rather  porous  type;  2.55  may  then  be  taken  as  the 
average  for  Mesozoic  and  Paleozoic  formations.  The  pre-Cambrian 

'"Redistribution  of  the  Elements  in  the  Formation  of  Sedimentary  Rocks," 
Jour.  Geol.,  XV  (1907),  238-56. 
3  Loc.  tit. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  215 

rocks  contain  both  igneous  and  sedimentary  formations,  but  the 
considerable  iron  ore  and  metamorphic  nature  would  bring  the 
specific  gravity  of  the  sediments  somewhat  above  the  average 
of  2 . 70  for  non-porous  sediments.  Broad  areas  of  pre-Cambrian 
probably  range  therefore  between  2 . 75  and  3 .00  in  specific  gravity. 
More  limited  areas,  because  of  a  predominance  of  granite  and 
quartzite,  may  range  as  low  as  2 . 70.  About  2 . 67,  however,  would 
be  a  minimum. 

As  these  are  merely  averages  it  is  better  in  basing  calculations 
upon  them  to  assume  a  certain  range  in  density  for  each  figure  and 
to  obtain  thus  a  knowledge  of  the  influence  of  reasonable  variations 
upon  the  results.  The  data  may  then  be  tabulated  as  follows: 

TABLE  IX 
ESTIMATED  MEAN  SPECIFIC  GRAVITIES  OF  GEOLOGIC  FORMATIONS 

Pre-Cambrian 2 .  75-2 . 80 

Paleozoic  and  Mesozoic 2 .  50-2 . 60 

Cenozoic 2 . 40-2 .  50 

The  range  in  these  specific  gravities  shows  the  necessity  of  con- 
sidering them  in  all  refined  calculations  on  the  anomalies  of  gravity. 
In  place,  however,  of  using  a  mean  density  figure  for  all  stations  on 
formations  of  a  certain  geologic  age,  it  would  be  of  much  more 
value  to  have  measurements  of  the  actual  surface  densities  occurring 
in  each  area;  also  estimates  by  geologists,  based  on  geologic 
structure  and  these  surface  measurements,  of  the  densities  extend- 
ing to  the  base  of  the  sedimentary  rocks  of  each  locality. 

It  seems  probable  from  the  mean  density  of  2 . 80  obtained  for 
igneous  rocks  that  the  density  of  2.67  used  by  geodesists  for  the 
mean  density  of  the  zone  of  compensation  is  too  low.  If  any 
variation  from  the  average  composition  takes  place  with  depth 
within  the  limits  of  76  miles,  it  is  likely  to  be  a  variation  toward 
more  basic  and  heavier  rocks.  Assuming,  however,  an  average 
uniformity  of  chemical  composition,  the  opposing  effects  of  tem- 
perature and  pressure  remain  to  be  considered.  Using  the  coeffi- 
cient of  expansion  of  the  average  igneous  rock  computed  by  W.  H. 
Emmons,1  0.000,019,9  for  i°C.,  and  a  temperature  gradient  of 

1  Chamberlin  and  Salisbury,  Geology,  I  (1904),  547. 


2i6  JOSEPH  BARRELL 

i°  F.  for  60  ft.  in  depth,  gives  an  aggregate  expansion  of  3 . 6  per 
cent  to  the  outer  76  miles.  Using  6,500,000  as  the  modulus  of 
cubic  compressibility  of  the  average  rock  in  pound-inch  units1 
gives  a  total  compression  of  3 . 7  per  cent  to  the  outer  76  miles  due 
to  pressure;  that  is,  the  volume  effects  of  heat  and  pressure  prac- 
tically offset  each  other  within  the  zone  of  isostatic  compensation. 
Therefore  2 . 80  appears  to  be  the  lowest  mean  figure  which  should 
be  taken.  The  use  of  2 . 67  as  a  mean  figure  requires  for  isostatic 
equilibrium  a  density  of  but  2.60  extending  to  a  depth  of  76  miles 
under  land  3  km.  high,  a  figure  lower  than  the  specific  gravity  of 
granite. 

INTERPRETATION   OF  ANOMALIES  IN  TERMS   OF  MASS   AND  DEPTH 

Suppose  that  the  zone  of  isostatic  compensation  is  not  ol 
uniform  density  under  any  one  station,  but  contains  masses  of 
variable  density  irregularly  distributed.  Let  these  masses  be  of 
considerable  thickness  and  area  as  compared  to  the  depth  of  the 
zone  of  compensation.  Suppose  that  the  topography  is  so  adjusted 
to  the  aggregate  density  that  the  pressures  are  everywhere  equal 
at  the  bottom  of  the  zone  of  compensation.  Abnormally  light 
masses  would  then  have  to  be  balanced  by  abnormally  heavy 
masses  in  the  same  column.  There  would  still  be  deflections  of 
the  vertical  and  anomalies  of  gravity  because  gravitation  varies 
inversely  with  the  square  of  the  distance,  the  upper  and  adjacent 
masses  of  abnormal  density  affecting  the  station  more  than  those 
more  distant  ones  of  opposite  abnormality  lying  vertically  below 
the  upper.  The  residuals  from  deflection  and  gravity  measure- 
ments would  under  such  an  arrangement  measure  strains  within 
the  outer  crust  but  not  upon  its  bottom.  The  strains,  if  produced 
by  abnormalities  in  the  upper  parts  of  the  crust,  would  further  be 
proportionately  smaller  and  yet  give  rise  to  residuals  of  a  certain 
magnitude  than  if  produced  by  abnormalities  in  the  lower  parts 
of  the  crust.  This  aspect  of  the  problem  must  be  investigated 
before  any  final  significance  regarding  the  strength  of  the  crust 
can  be  attached  to  the  grouping  of  residuals  discussed  under  the 

1  F.  D.  Adams  and  E.  G.  Coker,  An  Investigation  into  the  Elastic  Constants  of 
Rocks,  More  Especially  with  Reference  to  Cubic  Compressibility,  1906,  p.  67. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST 


217 


last  part  of  Part  II.     It  leads  to  a  consideration  of  the  relations 
between  mass,  distance,  and  anomaly. 

Under  the  title  of  "  Interpretation  of  Anomalies  in  Terms  of 
Masses  "*  Hayford  and  Bowie  show  that  the  excesses  and  deficiencies 
of  mass  to  a  great  distance  have  an  effect  upon  the  gravity  anomalies 
and  that  therefore  the  guarded  expression  "net  effective  excess 
(or  deficiency)  of  mass"  is  necessary  for  correctness.  They  give 
the  following  tabulation  to  show  the  influence  of  uncompensated 
masses  in  the  crust  in  giving  gravity  anomalies  when  the  gravity 
is  computed  on  the  assumption  of  isostasy:2 

TABLE  X 

Each  tabular  value  is  the  vertical  attraction  in  dynes  produced  at  a  station  by  a 
mass  equivalent  to  a  stratum  100  ft.  thick,  of  density  2 .67,  and  of  the  horizontal  extent 
indicated  in  the  left-hand  argument,  if  that  mass  is  uniformly  distributed  from  the  level 
of  the  station  down  to  the  depth  indicated  in  the  top  argument  and  from  the  station  in 
all  directions  horizontally  to  the  distance  indicated  in  the  left-hand  argument. 


DEPTH 

RADIUS  OF  MASS 

i.ooo  Ft. 

5,000  Ft. 

10,000  Ft. 

15,000  Ft. 

113.7  Km. 

i,28om.  (the  outer  radius  of 
zone  E)  

o  0029 

0.0018 

O.OOII 

0.0008 

o.oooo 

166.7  km.  (the  outer  radius  of 
zone  O)  

0.0037 

o  .  0034 

o  .  0034 

0.0034 

0.0024 

i,  190  km.  (or  io°4o',  the  outer 
radius  of  zone  10) 

o  0040 

o  0037 

o  0037 

o  0037 

o  0034. 

On  p.  in  it  is  concluded  by  these  authors  that  the  best  working 
hypothesis  is  to  take 

each  o .  0030  dyne  of  anomaly  as  due  to  an  excess  (or  deficiency)  of  mass  equiva- 
lent to  a  stratum  100  ft.  thick.  This  working  hypothesis  is  equivalent,  as 
may  be  seen  by  inspection  of  the  table  just  given,  either  to  the  assumption 
that  the  excess  (or  deficiency)  of  mass  is  uniformly  distributed  to  a  depth  of 
113. 7  kilometers  and  extends  to  a  distance  of  more  than  166.  7  kilometers  and 
less  than  1,190  kilometers  from  the  station,  or  that  it  extends  to  a  distance  of 
166. 7  kilometers  from  the  station  and  is  distributed  to  an  effective  mean  depth 
of  more  than  1 5,000  feet  and  less  than  113.7  kilometers,  or  the  working  hypothe- 
sis may  be  considered  to  be  a  combination  of  these  two  assumptions. 

The  mean  anomaly  of  0.018  dyne,  interpreted  on  this  basis  of 
0.030  dyne  being  taken  as  equivalent  to  100  ft.  of  mass,  gives  a 


Hayford  and  Bowie,  p.  108. 


'Ibid.,  1912,  p.  109. 


2i8  JOSEPH  BARRELL 

mean  departure  from  isostatic  compensation  amounting  to  600  ft.; 
given  more  exactly  by  Bowie  as  630  ft. 

It  is  seen  from  the  quoted  statement  that  the  authors  accept, 
first,  as  one  alternative  a  very  widespread  regional  net  excess  (or 
deficiency)  of  mass  uniformly  distributed  in  depth;  or,  second,  a 
somewhat  broad  regional  distribution  but  confined  to  the  outer 
part  of  the  zone  of  compensation;  or,  third,  some  combination  of 
the  two  assumptions. 

The  first  assumption  would  throw  a  real  strain  upon  the  bottom 
of  the  zone  of  compensation  and  signifies  regional  compensation  to 
limits  very  far  beyond  those  stated  elsewhere  by  the  authors.  It 
is  therefore  inconsistent  from  that  standpoint,  but  gives  a  smaller 
vertical  load  and  consequently  a  smaller  vertical  departure  from 
the  level  giving  isostatic  equilibrium  than  would  a  more  limited 
area.  If,  for  example,  it  be  assumed  that  the  radius  of  the  zone 
limiting  regional  compensation  is  58.8km.,  which  is  about  the 
maximum  limit  for  regional  compensation  which  Hayford  allows 
elsewhere;  then  it  may  be  computed  that  for  uniform  distribution 
of  the  excess  (or  deficiency)  of  mass  to  a  depth  of  114  km.,  a  mass 
equivalent  to  100  ft.  of  density  2.67  corresponds  to  an  anomaly 
of  but  0.0013  dyne  instead  of  0.0030.  This  would,  for  a  mean 
anomaly  of  0.018,  signify  an  average  departure  over  the  United 
States  of  1,380  ft.  from  the  level  giving  isostatic  equilibrium,  instead 
of  600  ft. 

The  second  assumption,  that  the  excess  (or  deficiency)  is  in  the 
outer  part  of  the  crust,  gives  also  a  much  higher  anomaly  for  a 
unit  mass  than  would  an  equally  permissible  assumption  that  the 
excesses  or  deficiencies  occurred  at  various  levels  and  on  the  average 
were  at  a  depth  of  one- third  or  one-half  of  the  zone  of  compensation. 
The  relationship  of  anomalies  to  geologic  formations,  to  be  dis- 
cussed later,  shows  certain  variations  in  density  in  the  outer  crust, 
but  the  greater  parts  of  the  anomalies  are  not  due  to  this  cause. 
From  the  previous  discussion  on  the  limits  of  regional  compensa- 
tion it  would  seem  that,  on  the  assumption  that  the  excesses  or 
deficiencies  of  mass  are  on  the  whole  uniformly  distributed,  0.0024 
would  be  an  appropriate  figure  to  use  as  the  mean  anomaly  for 
unit  thickness  of  mass.  The  highest  anomalies,  however,  are 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  219 

probably  better  interpreted  by  o .  0030  as  a  divisor,  since  as  a  class 
they  must  be  assumed  as  due  to  excesses  or  deficiencies  of  mass 
which  are  both  near  and  large.  This  does  not  mean,  however,  that 
the  larger  masses  are  not  assumed  as  scattered  uniformly,  according 
to  the  laws  of  chance,  through  the  crust.  It  is  seen,  then,  that 
Hayford  and  Bowie  have  favored  those  interpretations  which  gave 
a  large  anomaly  per  unit  mass  and  have  ascribed  the  total  anomaly 
as  on  the  average  to  be  interpreted  on  this  basis,  obtaining  there- 
by a  smaller  figure  as  the  mean  departure  in  feet  from  the  level 
for  perfect  compensation.  They  have  not  discussed,  furthermore, 
in  the  text  the  influence  of  deeper-seated  variations  of  density, 
which  might  give  considerable  residuals,  nor  the  possibility  that 
departures  from  the  mean  density  in  opposite  directions  might 
balance  each  other  so  as  to  give  equal  pressures  at  the  bottom  of  the 
zone  of  compensation.  The  latter  case  will  not  seem  improbable 
to  the  geologist.  The  great  batholiths  of  the  Archean  appear  to 
make  a  universal  floor  in  the  crust.  They  range  in  composition 
from  granites  to  gabbros  and  have  come  to  rest  at  various  levels. 
Light  and  heavy  masses  may  well  be  irregularly  distributed  in 
the  same  vertical  cylinder.  If  at  the  time  of  origin  the  whole 
were  too  heavy,  a  tendency  would  have  arisen  for  the  column  to 
sink  until  equilibrium  was  attained.  If  the  whole,  on  the  con- 
trary, were  too  light,  the  column  would  have  tended  to  rise  until 
a  heavier  base  balanced  the  lighter  mass  above.  Thus,  if  irregular 
distribution  of  density  arose  as  the  result  of  vertical  igneous 
intrusion,  the  whole  region  would  tend  to  seek  that  level  where 
the  irregularities  would  balance. 

In  order  to  gain  quantitative  ideas  as  to  this  possibility  of 
partly  explaining  the  anomalies,  the  writer  has  made  calculations 
on  the  following  assumptions.  A  station  is  situated  upon  the  axis 
of  a  vertical  cylinder  extending  from  the  station  to  a  depth  of 
1 14  km.  The  radius  is  taken  successively  at -58. 8,  166.7,  and 
1,190  km.  Let  such  a  cylinder  be  divided  into  five  equal  cylinders 
by  horizontal  planes.  Let  each  of  the  five  be  equivalent  in  mass 
to  a  cylinder  of  the  same  radius  but  only  100  ft.  in  depth  and  of 
density  2 . 67 ;  in  other  words,  the  unit  mass  as  used  by  Hayford 
and  Bowie.  What  will  be  the  attraction  in  dynes  per  gram  pro- 


22O 


JOSEPH  BARRELL 


duced  at  the  station  by  each  cylinder  respectively?1    The  results 

are  as  follows: 

TABLE  XI 

VERTICAL  ATTRACTION  IN  DYNES  ON  ONE  GRAM  AT  STATION  BY  CYLINDER  22.8  KM. 

THICK,  DENSITY  0.00357,  EQUIVALENT  IN  MASS  TO  THICKNESS 

OF  100  FT.  AT  DENSITY  2.67 


No.  of  Cylinder 

Depth  in  Km. 
from  Station  to 
Top  of  Cylinder 

Attraction  for 
Radius  of 
58.8  Km. 

Attraction  for 
Radius  of 
166.7  Km. 

Attraction  for 
Radius  of 
noo  Km. 

I  
II  
Ill  

O.o 
22.8 
4^  •  6 

0.0031 
0.0017 
O   OOIO 

0.0032 
O.OO28 
o  0024 

O  .  0036 
0.0035 
O   OO3< 

IV  

68.4 

o  0007 

O   OO2O 

o  003  ^ 

v  

QI  .  2 

o  0005 

o  0017 

O   OO34. 

The  results  for  radius  58.8  km.  show  that  masses  of  this  size 
situated  near  the  bottom  of  the  zone  of  compensation  exert  but  a 
fraction  of  the  influence  given  by  equivalent  masses  near  the  sur- 
face. A  balancing  of  light  and  heavy  masses  in  a  column  of  this 
radius  would  give  isostasy  at  the  base  and  yet  produce  notable 
anomalies.  For  radius  166.7  km.  the  importance  of  depth  is 
much  diminished.  For  radius  1,190  km.  it  practically  disappears. 
This  means  that  a  wide  regional  variation  in  depth  with  plus  and 
minus  departures  from  the  uniform  density,  the  light  and  heavy 
layers  balancing,  would  not  produce  anomalies  provided,  as  stated, 
there  was  isostatic  equilibrium  at  the  base. 

To  give  a  somewhat  extreme  illustration;  suppose  that  the 
upper  cylinder,  I,  is  2  per  cent  lighter  than  the  mean  density  of 

1  The  formula  for  making  these  computations  was  kindly  worked  out  for  me  by 
Professor  H.  S.  Uhler,  checking  it  as  given  by  B.  O.  Pierce,  Newtonian  Potential  Func- 
tion, p.  8.  It  is  as  follows: 


in  which 

F  =  force  in  dynes  per  gram. 

p  =  density,  in  this  case  =0.003,5  7. 

Y  =constant  of  gravitation  =  o  .000,000,066,58. 

o  =  radius  of  cylinder. 

c  =  distance  on  axis  from  station  to  top  of  cylinder. 

A=depth  of  cylinder;  in  this  case  22  .8  km. 

For  radii  of  58.8  and  166.7  km.  no  correction  need  be  made  for  curvature  of  the  earth's  surface. 
For  0  =  1190  km.  an  empirical  correction  was  obtained  by  comparing  the  results  with  Hayford's 
computations. 

The  writer  overlooked  until  later  the  fact  that  Hayford  and  Bowie  also  give  this 
formula  with  a  different  notation  on  p.  1  7  of  their  work. 


.  THE  STRENGTH  OF  THE  EARTH'S  CRUST 


221 


2.67  and  the  lower  cylinder,  V,  is  2  per  cent  heavier.  Let  these 
abnormalities  be  limited  areally  to  the  cylinder.  This  is  a  departure 
in  density  of  o.  054, 15.1  times  the  density  o .  00357.  The  anomalies 
will  be  as  follows: 

TABLE  XII 

ANOMALIES  DUE  TO  IRREGULAR  VERTICAL  DISTRIBUTION  OF  DENSITY 


DENSITY  2  PER 

ANOMALIES 

No.  oy  CYLINDER  FROM  TABLE 

CENT  FROM 
MEAN 

Radius 
58.  8  Km. 

Radius 
166.7  Km. 

Radius 
IIQO  Km. 

I.. 

2  616 

—  o  047 

—  o  048 

—  o  0^4. 

V.  

2.  724 

+o  008 

-j-o  026 

H-O  051 

Resultant  anomaly 

—  o   O3Q 

—  O  O22 

—  o  003 

It  is  seen  from  this  tabulation  that,  first,  irregular  superposed 
but  balanced  positive  and  negative  distributions  of  density  up  to 
distances  as  large  as  the  radii  of  the  areas  of  grouped  residuals 
could  produce  at  least  a  considerable  part  of  the  anomalies;  or, 
second,  actual  departures  from  isostatic  equilibrium  with  the 
resultant  strain  on  the  crust  could  produce  them;  or,  third,  a 
combination  of  the  two.  In  the  second  case,  as  Hay  ford  and 
Bowie  show,1  the  anomalies  could  result  from  a  layer  a  few  miles 
thick  adjacent  to  the  station  and  of  very  abnormal  density;  or 
from  deep  and  regional  masses  of  great  volume,  but  departing 
only  slightly  from  the  mean  density.  The  choice  between  these 
several  alternatives,  or  the  degree  to  which  they  co-operate,  must 
be  investigated  under  the  following  topics. 

RELATIONS   OF  ANOMALIES  TO  EXPOSED   GEOLOGIC  FORMATIONS 

The  latest  data  given  by  Bowie  on  this  subject  are  shown  in 
Table  XIII  (p.  222):2 

These  figures  of  course  are  not  to  be  regarded  as  of  high  pre- 
cision, as  may  be  seen  by  comparing  the  earlier  and  later  results. 

*0p.cit.,  Pp.  108-11. 

2 "  Some  Relations  between  Gravity  Anomalies  and  the  Geologic  Formations  in 
the  United  States,"  Am.  Jour.  Sci.  (4),  XXXIII  (1912),  237-40. 


222 


JOSEPH  BARRELL 


Hayford  and  Bowie  in  their  successive  publications  give  the  follow- 
ing for  the  pre-Cambrian  and  Cenozoic  stations,  the  two  groups 


TABLE  XIII 


Geologic  Formation 

Number  of  Stations 

Mean  with 
Regard  to  Sign 

Mean  without 
Regard  to  Sign 

Pre-Cambrian 

IO 

-J-O  016 

o  026 

Paleozoic 

21 

—  o  003 

O   OIO 

Mesozoic   

20 

-j-o  002 

O   OI  ? 

Cenozoic          .  . 

20 

—  o  008 

O   O2I 

Intrusive  and  Effusive 
Unclassified.  . 

II 

22 

—  0.007 

-J-O   OI  I 

0.015 

O   O2O 

All  stations  

123 

o  ooo 

O   OIO 

to  which  the  attention  will  be  confined.  A  few  stations  of  high 
anomaly  must  have  considerable  influence  on  the  result,  as  most 
of  the  stations  are  used  in  common  in  all  of  the  estimates. 

TABLE  XIV 


Geologic 
Formation 

Number  of 
Stations 

Mean  with 
Regard  to  Sign 

Mean  without 
Regard  to  Sign 

Hayford  and  Bowie,  U.S.C.J 

Pre-Cambrian 

7 

+0.019 

0.026 

and  G.S  \ 

Cenozoic 

20 

—  O  Oil 

O   O2I 

Bowie,  U.S.C.  and  G.S  { 

Pre-Cambrian 
Cenozoic 

& 

+0.024 
—  0.007 

0.024 

O.O2I 

Bowie,  Am.  Jour.  Scl  < 

Pre-Cambrian 
Cenozoic 

IO 

29 

+0.016 
—0.008 

O.026 
O.O2I 

*  Fifteen  stations  have  plus  anomalies,  1 7  have  minus  anomalies. 

Bowie's  figures  in  the  American  Journal  of  Science  will  be  used  in 
the  following  discussion. 

Bowie  favors  the  explanation  that  these  relations  of  anomalies 
to  geologic  formations  are  due  to  slight  changes  of  density  extend- 
ing more  or  less  through  the  zone  of  compensation  and  leading  to 
departures  from  perfect  isostasy.  The  writer,  however,  is  led  to 
favor  the  view  that  about  one-half  of  the  contrasted  anomaly  for 
these  two  groups  is  due  to  a  lesser  density  within  the  outer  mile 
of  crust  beneath  the  Cenozoic  stations,  as  contrasted  to  the  outer 
mile  of  crust  beneath  the  pre-Cambrian  stations.  The  remainder 
of  the  anomaly  it  is  thought  is  explained  by  the  ease  of  erosion  of 
Cenozoic  formations,  the  resistance  to  erosion  of  the  pre-Cambrian 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  223 

rocks.  The  latter  consequently  tend  to  stand  above  the  regional 
levels.  They  therefore  possess  surficial  excess  both  of  density 
and  volume. 

The  average  thickness  of  sedimentary  rocks  if  spread  uniformly 
over  the  globe  is  thought  to  be  between  2,000  and  2,500  ft.1  Over 
the  pre-Cambrian  areas  it  must  average  much  less;  over  the  areas 
of  later  formations  much  more.  Under  the  Cenozoic  stations 
assume: 

i  ,000  ft.  of  sediments  at  density 2 . 40  to  2 . 50 

4,000  ft.  of  sediments  at  density 2 .  50  to  2 . 60 

Giving  a  total  of  5,000  ft.  at  density 2 . 48  to  2 .  58 

With  a  deficiency  of  density  of o.  19  to  o. 09 

Under  the  pre-Cambrian  stations  assume: 

5,000  ft.  of  crystalline  rock  at  density 2 . 75  to  2 . 80 

An  excess  of  density  of o. 08  to  o.  13 

This  does  not  involve  the  improbable  assumption  that  below  the 
outer  5,000  feet  of  crystalline  rock  of  density  2 . 75  to  2 . 80  the  den- 
sity suddenly  decreases  to  2.67  and  then  remains  constant  through- 
out the  zone  of  compensation.  The  vertical  density  gradient,  if 
uniform  for  all  points,  has  but  little  effect,  it  being  the  horizontal 
variations  of  density  which  enter  into  the  problem  of  isostasy.  To 
maintain  conformity  with  Hayford's  figures,  therefore,  the  density 
2.67  will  be  frequently  assumed  as  the  mean  density  of  the  litho- 
sphere,  although  the  previous  discussion  shows  that  it  cannot  be 
assumed  as  the  density  of  the  outer  mile  of  crystalline  rocks  when 
comparing  these  to  the  mile  of  sedimentary  rocks  taken  as  the  mean 
depth  underlying  the  Cenozoic  stations. 

In  comparison  with  this  thickness  of  5,000  ft.  the  average  area 
of  formations  is  very  great.  A  plane  sheet  of  rock  100  ft.  thick  and 
of  density  2.67,  if  of  indefinite  extent,  will  produce  an  anomaly  of 
0.0034  dyne  upon  a  point  outside  of  it,  irrespective  of  the  distance 
to  that  point.  This  theory  may  be  applied  without  gross  error 
to  the  relation  of  surface  geologic  formations  to  anomalies.  If 
this  unit  mass  be  expanded  from  100  to  5,000  ft.  thickness,  the 

1  F.  W.  Clark,  "Data  of  Geochemistry,"  Bull  491,  U.S.  Geol.  Surv.,  1911,  p.  30. 


224  JOSEPH  BARRELL 

density  will  be  decreased  to  0.053  that  of  water.     The  data  may 
then  be  tabulated  as  follows: 

TABLE  XV 

COMPUTED  ANOMALIES  DUE  TO  DENSITIES  OF  SURFACE  FORMATIONS 


Deficiencies  or 
Excesses  of  Density 

Anomalies  in  Dynes 
per  Gram  Due  to 
Thickness  of  5,000  Ft. 

Unit  mass         .... 

O  CK"? 

O  0034 

Cenozoic.         

—  O   IQ 

—  O   OI2 

Pre-Cambrian  

—  0.09 

+0.08 

—  0.006 

-f-o  005 

+0.13 

+0.008 

These  mean  anomalies  of  the  pre-Cambrian  due  to  the  greater 
density  of  the  outer  5,000  ft.  of  rock,  when  compared  to  the  Cenozoic 
anomalies,  are,  as  shown  by  this  tabulation,  at  a  minimum  o.on 
greater,  at  a  maximum  0.020  greater,  at  a  mean  0.0155  greater. 
The  difference  of  the  means  shown  by  geodetic  measurement  was 
0.024.  The  specific  gravities  seem  to  have  been  taken  as  far 
apart  in  limits  as  is  allowable  and  the  assumed  mean  thickness  of 
sediments  as  5,000  ft.  beneath  the  Cenozoic  stations  is  a  generous 
figure;  the  mean  thickness  is  more  likely  to  be  less,  rather  than 
greater.  The  means  for  the  geodetic  anomalies  as  related  to 
geologic  formations  are  perhaps  subject  to  about  the  same  degree 
of  error  as  the  determinations  of  the  anomalies  from  the  specific 
gravities  and  thickness.  The  result,  although  not  of  a  high  order 
of  accuracy,  shows  that  although  the  range  in  specific  gravities 
accounts  for  a  considerable  part,  perhaps  one-half  or  two-thirds, 
of  the  relation  of  anomalies  to  geologic  formations,  it  can  hardly 
account  for  the  whole. 

To  find  the  cause  for  the  remaining  portion  of  the  anomaly,  two 
hypotheses  may  be  considered:  first,  that  it  is  due  to  a  slight 
regional  excess  of  density  extending  to  a  depth  of  114  km.,  the 
hypothesis  favored  by  Bowie;  or,  second,  that  the  Archean  areas 
on  the  average  stand  higher  than  the  Cenozoic  by  virtue  of  resistance 
to  erosion. 

The  geologic  evidence  as  it  is  at  present  understood  is  against 
the  first  hypothesis  and  in  favor  of  the  second.  This  statement 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  225 

is  based  on  the  view  that  Archean  and  Proterozoic  areas  have 
tended  to  be  rising  elements  of  the  continent.  Erosion  instead  of 
sedimentation  has  been  dominant  in  later  geologic  time,  which  is 
the  reason  why  these  rocks  are  now  exposed  as  surface  formations. 
If  there  is  any  deep-seated  departure  of  density  from  the  mean 
this  tendency  to  rise  should  correspond,  however,  to  a  deficiency 
of  density  persisting  through  the  geologic  ages,  extending  through 
much  of  .the  zone  of  compensation  and  offsetting  the  more  than 
average  surface  density.  Such  a  regional  deficiency  is  opposite 
in  character  to  the  excess  which  is  postulated  by  Bowie  as  an 
explanation  of  the  positive  anomalies. 

Assume  then  as  the  next  step  in  the  argument  that  the  density 
of  the  zone  of  compensation  beneath  the  pre-Cambrian  areas  to  a 
depth  of  114  km.  is  the  same  as  under  Cenozoic  areas  except  for  the 
outer  5,000  ft.,  both  having  a  mean  density  of  2.75  to  2.80,  but 
taken  here  as  2.67.  The  outstanding  anomaly  in  that  case  is  due 
to  a  longer  mean  column  for  the  pre-Cambrian  areas  and  conse- 
quently greater  mass  above  the  level  of  complete  compensation. 
If  the  mean  radius  of  these  longer  pre-Cambrian  and  shorter  Ceno- 
zoic columns  is  as  great  as  166.7  km.,  then  the  unit  excess  or 
deficiency  of  mass  of  100  ft.  at  density  2.67  when  spread  over  these 
columns  will  correspond  to  an  anomaly  of  0.0024.  If  the  mean 
effective  areas  of  the  pre-Cambrian  and  Cenozoic  formations 
affecting  individual  stations  are  less,  the  unit  mass  will  give  a 
smaller  unit  anomaly.  If  the  mean  effective  areas  are  greater,  the 
unit  anomaly  will  not,  however,  rise  above  0.0035.  Assume  then 
in  conclusion  a  mean  radius  of  166.7  km.,  an  anomaly  of  0.0024 
dyne  as  resulting  from  100  ft.  of  added  mass  of  mean  density,  and 
the  outstanding  anomaly  not  accounted  for  by  the  surficial  densities 
but  due  to  an  outstanding  difference  in  volume  as  between  0.008 
and  0.012.  These  figures  correspond  to  a  differential  mean 
elevation  of  330  to  500  feet  of  the  pre-Cambrian  above  the  Cenozoic, 
due  to  erosion.  To  physiographers  such  a  conclusion  will  seem 
quite  in  accord  with  the  geologic  evidence  testifying  to  the  resistance 
of  pre-Cambrian  formations. 

The  character  of  the  Archean  and  Proterozoic  anomalies  enters 
into  the  problem  of  crustal  rigidity  in  the  following  way.  If  there 


226  JOSEPH  BARRELL 

were  local  and  close  compensation,  then  as  erosion  removed  the 
softer  surrounding  rocks  there  should  be  isostatic  upwarping  of  such 
areas  of  denudation  and  relative  downwarping  of  the  uneroded 
crystalline  areas.  Such  warping  of  the  Mohawk,  St.  Lawrence, 
and  Champlain  valleys  with  respect  to  the  Adirondacks  has  not 
been  noted,  though  the  problem  from  the  standpoint  of  field 
evidence  has  not  been  fully  studied.  The  physiographic  evidence 
that  residual  mountain  masses  known  as  monadnocks  or  unakas 
have  not  been  shown,  however,  to  be  marked  by  local  downwarping 
and,  on  the  contrary,  certainly  stand  in  relief  due  to  circumdenuda- 
tion,  combines  with  the  geodetic  evidence  of  the  average  excess 
of  gravity  for  the  resistant  areas  of  pre-Cambrian  formations,  to 
suggest  effective  rigidity  against  the  stresses  produced  by  erosion. 
The  evidence,  however,  as  developed  thus  far  from  the  geodetic 
standpoint  shows  that  there  are  more  important  factors  than  that 
of  the  surface  geologic  formation,  since  the  larger  anomalies  are 
much  greater  than  these  figures  which  have  been  discussed  and 
hold  but  little  relation  to  either  relief  or  surface  geology.  In  fact 
Hayford  and  Bowie  do  not  find  any  discoverable  relation  between 
the  anomalies  in  general  and  the  topography. 

It  is  thought  by  the  writer,  however,  that  if  stations  were 
located  especially  to  test  the  intensity  of  gravity  over  various 
broad  plateaus  remaining  by  circumdenudation  and  the  intensity 
compared  with  that  over  adjacent  broad  areas  of  lower  level,  the 
mean  differential  anomalies  due  to  the  surface  excess  of  mass  in 
the  plateau  over  the  lowlands  would  rise  to  a  larger  figure  than 
the  0.008  to  0.012  dyne  which  has  remained  to  be  explained  in 
the  present  discussion.  These  figures  are  low  because  certain 
pre-Cambrian  areas,  like  those  in  the  vicinity  of  Baltimore  and 
Washington,  have  been  lowered  by  prolonged  denudation  and  do 
not  stand  markedly  above  the  level  of  younger  formations.  Further- 
more, the  tendency  of  broad  pre-Cambrian  areas  to  stand  above 
sea-level  is  very  probably  of  an  isostatic  nature.  This  implies 
under  such  areas  a  slightly  lower  mean  density  to  the  whole  zone 
of  compensation  which  would  diminish  the  anomaly  due  to  the 
surface  elevation.  In  individual  areas  of  100  to  200  km.  radius, 
however,  such  a  relation  of  positive  anomaly  to  pre-Cambrian 


THE  STRENGTH  OF  THE  EARTH'S  CRUST 


227 


formations  and  plateaus  of  circumdenudation  may  not  be  found, 
since  it  is  clear  that  the  anomaly  from  this  cause  may  be  much 
more  than  neutralized  by  other  causes.  A  large  number  of  stations 
covering  broad  areas  would  therefore  be  required  adequately  to 
eliminate  these  other  influences  from  the  means. 

LARGE   OUTSTANDING  ANOMALIES   NOT   RELATED   TO   GEOLOGY   OR 

TOPOGRAPHY 

In  Fig.  5,  of  Part  II,  the  anomalies  are  shown  for  all  stations 
in  the  United  States.  It  is  seen  that  they  possess  an  areal  grada- 
tion in  magnitude  which  permits  the  drawing  of  anomaly  contours. 
The  excessive  anomalies  of  both  signs  cover  oval  areas  in  various 
parts  of  the  country  and  show  a  common  disregard  of  physio- 
graphic provinces,  structural  provinces,  and  geologic  formations. 
Looking  at  Fig.  5,  one  cannot  see  in  either  the  distribution  of 
anomalies  or  trends  of  contours  a  reflection  of  Atlantic  Coastal 
Plain,  or  Appalachian  Mountains,  or  Mississippi  Valley. 

Typical  examples  of  the  lack  of  necessary  relation  of  the  large 
anomalies  to  geologic  formations  are  seen  in  the  following  tabula- 
tion: 

TABLE  XVI 


No. 

Station 

Geologic  Formation 

Anomaly 

123  
74  
96  

Albany,  N.Y  
St.  Paul,  Minn  
Mena,  Ark  

Cambro-Ordovician 
Cambro-Ordovician 
Pennsylvanian    .  . 

-0.043 
+0.059 

—  O    O?2 

IOI  

Helen  wood,  Tenn.  . 

Pennsylvanian  .... 

~f~O    O4.O 

c-2     e;6 

Seattle  Wash 

Quaternary 

—  O   OO3. 

112 

Olympia   Wash 

Quaternary 

-L-O   OT.'i 

The  lack  of  relation  of  these  anomalies  to  topography  is  equally 
striking.  It  is  clear  then  that  internal  conditions  in  the  crust,  not 
expressed  on  its  surface,  must  be  the  principal  cause  of  these  larger 
departures  from  isostasy.  The  large  anomalies  show  their  relation- 
ship to  internal  causes  most  clearly,  but  the  smaller  anomalies  may 
also  by  analogy  be  ascribed  in  part  to  such  hidden  causes.  The 
results,  however,  of  surface  activities — -circumdenudation,  sedi- 
mentation, tangential  pressure,  or  extravasation — must  show  in 
large  ratio  over  regions  where  the  internal  variations  from  uniform 
density  are  small;  but  over  the  greater  part  of  the  United  States 


228  JOSEPH  BARRELL 

the  distribution  of  anomalies  appears  to  depend  more  upon  the 
internal  than  upon  the  external  departures  from  regional  uni- 
formity and  complete  isostasy.  The  internal  heterogeneities  of 
mass  are  therefore  presumably  greater  than  the  shiftings  of  mass 
due  to  external  activities. 

CRITERIA  FOR  SEPARATING  VERTICALLY  IRREGULAR  COMPENSATION 
FROM  REGIONALLY  INCOMPLETE   COMPENSATION 

Suppose  the  topography  smoothed  out  to  a  mean  level  over 
areas  as  large  as  the  limits  for  regional  isostasy.  The  deflection 
residuals  and  gravity  anomalies  would  then  be  due  to  one  or  more 
of  three  internal  causes;  first,  vertically  irregular  or  laterally 
displaced  compensation;  second,  regionally  incomplete  compensa- 
tion above  the  bottom  of  the  zone  of  compensation  because  of  the 
effective  rigidity  of  the  crust  above  that  level;  third,  regionally 
incomplete  compensation  above  a  certain  level  because  the  zone 
of  compensation  may  be  deeper  in  places,  transferring  stresses 
into  a  deeper  rigid  earth.  The  existence  of  a  general  approach 
toward  compensation  and  away  from  absolute  rigidity  suggests 
that  the  last  is  not  so  important  as  the  first  two  causes.  Under 
this  section  then  will  be  considered  these  two  causes,  their  effects 
upon  the  deflections  of  the  vertical  and  the  intensity  of  gravity, 
with  the  purpose  of  drawing  criteria  by  which  the  action  of  the 
two  causes  may  be  recognized  and  separated.  To  do  this  it  will  be 
necessary  to  discuss  here  to  some  extent  the  theory  of  the  attraction 
of  underground  masses  upon  stations  at  the  surface  of  the  earth. 
It  has  been  shown  that  balanced  irregularities  in  the  vertical 
distribution  of  densities  through  the  zone  of  compensation  could 
give  pronounced  anomalies  without  disturbing  the  isostatic  equilib- 
rium at  the  bottom  of  the  zone,  since  the  total  weight  of  the  column 
could  still  be  normal.  To  show  the  effect  of  such  balanced  irregu- 
larities upon  a  point  outside  of  the  column: 

Take  a  vertical  line  and  a  horizontal  line  which  intersect.  The 
masses  whose  effects  are  to  be  investigated  will  be  distributed  on 
the  vertical  line.  The  effects  are  to  be  determined  for  points  on 
the  horizontal  line.  To  express  the  trigonometric  relations  between 
any  point  on  the  vertical  and  any  point  on  the  horizontal  line,  let 
a  point  on  the  vertical  line  at  depth  D  be  defined  as  at  a  vertical 


THE  STRENGTH  OF  THE  EARTH'S  CRUST 


229 


angle  6  below  a  point  on  the  horizontal  line;  the  latter  to  be 
denned  as  at  distance  R  from  the  intersection. 

Let  the  gravitative  attraction  of  unit  masses  along  this  vertical 
line  upon  any  other  point  either  in  or  outside  of  this  line  be  repre- 
sented by  F.  The  horizontal  component  will  be  the  force  produ- 
cing deflection  of  the  vertical  and  may  be  represented  by  Fh.  The 
vertical  component  will  give  the  acceleration  of  gravity  due  to  the 
unit  mass  and  may  be  represented  by  Fv.  Taking  the  unit  mass 
such  that  the  constants  will  have  a  value  of  unity,  the  following 
relations  are  deduced: 

Attraction  of  unit  mass  at  depth  D,  upon  a  point  at  R: 

VL     cos3  e 
»— g- 

=  tan  0  cos3  0 

R2 
For  the  intersection  point, 

R  and  0=O  and 
Fk=o 


Let  the  depth  of  the  zone  of  compensation,  114  km.,  be  taken  as 
unit  distance,  i .  oo,  and  for  purposes  of  discussion  let  points  I,  II, 
III,  IV  be  located  on  a  vertical  line  at  depth  of  0.25,  0.50,  0.75, 
and  i. oo  as  shown  on  Fig.  6.  Solving  the  equations  for  these 
points  and  for  various  values  of  R  gives  the  following  tabulation: 

TABLE  XVII 

TABLE  OF  RELATIVE  ATTRACTIONS 
(Not  in  dynes  per  gram) 


ATTRACTION  BY  UNIT  MASSES  AT 

ATTRACTION  AT  STATIONS  FOR  VARIOUS  VALUES  OP  R 

No. 

Depth 

Angle 
below 

/?  =  I.OO 

R=o 

R=0.2$ 

R=o.$o 

R  =  i.oo 

£  =  2.00 

Fh 

Fv 

Fh 

Fv 

Fh 

Fv 

Fh 

Fv 

Fh 

Fv 

o  

0 

0.25 
0.50 
0-75 

I.OO 

O 
I4°02' 

26°34' 
36°52' 
45  °° 

o 
o 
o 
o 
o 

O 

16.00 
4.00 
1.78 

I.OO 

16.00 
S.6o 
1.44 
o-5i 

O.  21 

o 
S.6o 
2.88 
1-52 
0.91 

4.00 
2.88 
1.40 
0.68 
0.36 

o 
1.44 
1.40 
1.04 
0.72 

I.OO 

0.91 
0.72 
0-51 
o-35 

o 
0.23 
0.36 
0.38 
0-35 

0.25 
o.  24 
0.23 

O.2I 

0.18 

o 
0.03 
0.06 

0.08 
0.09 

I  

II  

Ill  
IV  

230 


JOSEPH  BARRELL 


Fig.  6  shows  the  curves  for  R=i.  For  any  other  value  of  R 
the  curves  would  be  the  same  in  form,  but  the  scales  of  ordinates 
and  abscissas  would  be  changed.  These  curves  may  be  used 
therefore  in  a  general  way. 


Attraction  on  A  by  points  on  vertical  line, 
shoam  by  abscissas  on  the  vertical  line. 
Horizontal         Vertical 
component  F>     component  Fv 

R^2S       R--.50  R.LOO 


Combined  attractions  of  I  and  IH  upon  points  on  the 
horizontal  line.shoiun  by  ordinates  on  the  horizontal  line 


Scale  of  distance. 
t.oo 


Horizontal  component 
Sum  of  I  and  HI 
Diff.  of  1  and  IH 

Vertical  component  f- 
Sum  of  I  and  HI 
Diff.  of  I  and  1U 


FIG.  6 


FIG.  7 


FIG.  6. — -Curves  showing  relative  attraction  of  all  points  on  the  vertical  line  upon 
a  point  at  distance  R=i. 

FIG.  7. — Combined  attractions  upon  all  points  on  the  surface  by  unit  masses  of 
like  and  unlike  signs  at  I  and  III  of  Fig.  6. 

The  table  shows  that  if  unit  masses  at  II  and  III  have  the  same 
sign  the  horizontal  component,  Fh,  for  the  sum  of  their  attractions 
at  o.  2$R  will  be  i  .95,  at  R  it  will  be  i .  23,  which  is  63  per  cent  of 
the  value  at  0.25^.  If  the  unit  masses  have  unlike  signs  the 
horizontal  component  of  their  difference  at  o.2$R  will  be  0.93,  at  R 
it  will  be  0.21,  which  is  but  23  per  cent  of  the  value  at  o.2$R. 
The  vertical  component,  Fv,  due  to  the  sum  of  the  masses  at  o.  25^ 
is  4.40;  at  .R  is  o. 74.  The  vertical  component  due  to  the  difference 
at  o.2$R  is  i  .35;  at  R  is  o. 02  and  of  opposite  sign.  It  is  noticed 
that  the  gravity  anomaly  diminishes  rapidly  with  increasing 
horizontal  distance  from  these  two  masses  and  passes  through 
zero.  The  deflection  of  the  vertical  first  increases  sharply  and 


THE  STRENGTH  OF  THE  EARTH'S  CRUST 


23 1 


then  diminishes,  but  less  rapidly  than  the  gravity  anomaly.  It  is 
important  to  notice  that  in  both  cases  the  total  influence  due  to 
masses  of  opposite  sign  diminishes  much  more  rapidly,  and  where 
their  distance  apart  is  0.25  their  influence  is  small  at  distance  R 
and  negligible  at  2R.  This  gives  a  means  of  determining  whether, 
in  the  crust,  anomalies  and  deflections  are  due  to  regional  departures 
from  isostasy  or  to  balanced  irregularities  in  density  without 
absence  of  isostasy  at  the  base  of  the  zone. 

To  give  a  further  illustration  of  balanced  departures  in  density 
spread  over  a  greater  vertical  distance,  and  representing  in  that 
way  perhaps  a  more  average  case,  assume  that  an  excess  or  deficiency 
equivalent  to  a  unit  mass  is  at  depth  0.25  and  another  at  depth 
0.75.  The  following  tabulation  shows  their  influence  upon  the 
surface  of  the  earth  at  increasing  horizontal  distances. 

TABLE  XVIII 

ATTRACTION  BY  UNIT  MASSES  AT  I  AND  III  UPON  POINTS  ON  THE  HORIZONTAL 

LINE 


Position 

Horizontal  Distance  on  Surface  of  Earth  from  Vertical  Line 

of  Mass 

o 

0.25 

0.50 

I  .OO 

2.OO 

4.00 

Fh  

HI 

O 

-6.  ii 

-3.56 

-1.42 

-0-55 

—  O.  121 

+m  f 

O 

-5-09 

—  2.  2O 

—  O.4O 

-0.03 

—  0.003 

Fv  

1     I 

-m  r 

-17.78 

-7.12 

-2.48 

—  0.61 

—  o.  ii 

—  .0.015 

+HI  / 

—  14.22 

-4.08 

—  O.4O 

+0.15- 

+0.05 

+0.007 

The  data  in  this  table  are  represented  by  the  curves  of  Fig.  7. 
It  shows  that  for  this  arrangement  of  masses  the  influence  on  the 
surface  falls  off  rapidly  at  a  horizontal  distance  between  0.25 
and  0.75,  which  are  also  the  vertical  depths  to  I  and  III.  When 
the  masses  are  of  opposite  sign  the  anomaly  passes  through  zero 
at  a  horizontal  distance  of  about  0.6,  and  the  deflection  force  for 
opposite  sign  decreases  to  half  the  value  of  the  sum  at  about  o.  75. 
The  ratio  between  the  effects  of  like  and  unlike  masses  becomes 
more  marked  the  greater  the  distance  of  the  point,  although  the 
actual  magnitudes  of  the  forces  decrease. 


. 


232  JOSEPH  BARRELL 

Now  assume  the  unit  masses  at  I  and  III  to  be  parts  of  masses 
of  like  density  extending  to  the  left  of  o  to  a  distance  N.  Consider 
the  aggregate  effect  upon  a  given  point,  as  that  at  0.50,  or  in 
general  at  point  R.  The  effect  of  each  unit  at  distance  x  to  the 
left  of  o  upon  the  point  at  o.  50  will  be  measured  by  an  ordinate  at 
a  distance  x  to  the  right  of  o .  50.  This  will  give  the  same  aggregate 
result  as  concentrating  the  masses  at  o  and  summing  up  the  area 
of  the  curve  to  the  right  of  the  point  at  o .  50  to  a  distance  of  o .  50+ 
N.  Stated  in  general  terms,  masses  at  depths  I  and  III  extending 
linearly  to  distance  N  to  the  left  of  o  will  have  an  aggregate  effect 
upon  a  point  R  equal  to  the  area  of  the  curve  between  R  and  R-{-N. 

As  to  the  aggregate  effect  on  Fv,  the  gravity  anomaly:  If  the 
two  sheets  are  of  negative  density,  it  is  seen  that  the  result  will 
be  an  increased  negative  anomaly  over  the  effect  of  the  separate 
unit  masses.  If  the  lower  mass  is,  however,  of  positive  density, 
the  result  for  ordinarily  limited  sheets  will  be  a  change  between 
o  and  0.50  from  a  large  negative  to  a  small  positive  anomaly. 
This  may  be  compared  with  the  effects  of  other  possible  distribu- 
tions of  mass  upon  the  gravity  anomaly. 

If  the  anomaly  due  to  the  adjacent  departure  from  uniform 
distribution  is  of  the  mean  value  or  greater,  the  more  distant  abnor- 
mal masses  will  have  but  relatively  small  influence.  This  is  because 
the  higher  anomalies,  with  the  exception  of  Seattle,  are  but  two 
or  three  times  the  mean.  Further,  in  a  zone  of  large  radius  there 
are  a  greater  number  of  positive  and  negative  departures.  Their 
aggregate  effect,  according  to  the  laws  of  chance  distribution  would 
increase  but  slowly  and  this  effect  is  diminished  by  distance  accord- 

Ffl=tan  0cos30 
ing  to  the  formula  -  ~2         — . 

A  reversal  from  a  large  anomaly  of  one  sign  to  a  large  anomaly 
of  opposite  sign,  rather  than  a  small  one  of  opposite  sign,  marks 
then  in  general  a  passage  from  an  area  of  excess  or  deficiency  of 
mass  to  the  opposite.  A  gradual  change  in  the  anomaly  is  the 
reflection  of  a  change  in  the  subsurface  abnormalities  nearly  as 
gradual.  If  the  areal  variations  show  that  the  passages  of  the 
anomaly  through  zero  are  not  frequent,  they  go  to  show  that 
limited  notable  irregularities  of  density  of  opposite  sign  in  the 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  233 

same  column  are  rare.  Furthermore,  it  has  been  shown  under 
the  topic  "The  Variable  Rate  of  Compensation  upon  Gravity 
Anomalies"  that  a  variable  distribution  of  balanced  densities 
has  more  effect  if  in  areas  of  between  100  and  200  km.  radius  and 
has  but  little  effect  on  anomalies  if  the  balanced  densities  extend 
over  much  larger  areas. 

As  to  the  aggregate  effects  produced  upon  Fh,  giving  deflection 
residuals,  by  these  sheets  I  and  III:  If  the  sheets  have  like  sign 
the  deflection  force,  as  shown  in  Fig.  7,  will  die  out  somewhat 
gradually  and  extend  to  considerable  distances.  If  they  have 
unlike  sign  the  deflection  force  will  fall  off  sharply  between  0.25 
and  i .  oo.  If,  however,  the  abnormalities  of  density  should  dis- 
appear gradually,  that  is,  if  the  sheets  did  not  terminate  sharply 
at  o,  this  rate  of  falling  off  would  be  slower.  Reversals  of  sign 
of  the  deflection  residuals  would  require  areal,  not  vertical,  irregu- 
larities of  mass.  They  could  not  take  place  as  an  effect  of  dis- 
tance from  a  single  mass  or  of  two  masses  of  unlike  sign  and  vertically 
over^each  other.  Where  sharp  reversals  of  sign  take  place  in  the 
deflection  residuals  the  presence  of  areally  contiguous  areas  of 
unlike  departures  in  mass  is  shown.  A  mere  difference  in  magni- 
tude of  excess  of  mass  but  of  the  same  sign  may,  however,  produce 
changes  in  the  sign  of  the  deflection  residuals.  In  the  irregular 
areal  distribution  of  abnormal  masses  not  balanced  by  being  over 
each  other,  the  deflection  areas  of  like  sign  would  thus  tend  to  be 
smaller  than  the  anomaly  areas  of  like  sign.  A  gradual  fading-out 
of  the  deflection  residuals  would  be  the  mark  of  gradual  fading-out 
of  the  abnormal  mass  or  the  increasing  influence  of  distant  masses. 

Various  special  combinations  of  three  or  more  masses  could  at 
any  one  point  simulate  the  relations  indicated,  but  such  special 
relations  would  not  be  of  common  occurrence  and  could  not  give 
a  generah'ty  of  relation  of  this  sort. 

There  have  thus  been  drawn  up  a  set  of  criteria  by  which 
balanced  irregularities  within  the  zone  of  compensation  may  be 
distinguished  from  regional  departures  from  isostasy.  It  remains 
to  apply  those  to  the  areal  distribution  of  gravity  anomalies  and 
deflection  residuals  as  given  by  Hayford  and  Bowie.  It  must  be 
recognized,  however,  that  the  stations,  although  numerous  as 


234  JOSEPH  BARRELL 

compared  to  previous  measurements,  are  yet  very  scattered  for 
the  precise  application  of  these  tests  and  can  at  best  give  but 
qualitative  results.  It  is  thought,  nevertheless,  that  the  general 
nature  of  the  answer  is  determinative. 

GRAVITY   ANOMALIES    CAUSED    LARGELY   BY    REGIONAL   DEPARTURES 

FROM  ISOSTASY 

The  first  question  is:  To  what  degree  do  the  areas  of  excess 
(or  deficiency)  of  mass  as  indicated  by  gravity  anomalies  coincide 
with  areas  of  excess  (or  deficiency)  as  shown  by  the  deflection 
residuals  ?  In  Fig.  51  there  are  indicated  a  number  of  ovals  shown 
in  dot-and-dash  outline  and  marked  +  or  — .  These  are  the 
definitely  bounded  areas  of  excess  or  deficiency  of  mass  indicated 
by  the  deflection  residuals.  The  entire  surface  of  the  crust  must 
be  constituted  of  such  areas,  but  only  a  few  are  surrounded  by 
sufficient  observations  to  permit  a  boundary  to  be  drawn  at  present. 
Even  this  boundary  must  not  be  regarded  as  sharply  definite. 
Beside  these  ovals  there  are  shown  in  illustrations  5  and  6,  Hayford, 
1909,  areas  of  residuals  characterized  by  like  sign,  referred  to  in 
the  present  paper  as  ''areas  of  grouped  residuals."  They  are  not 
definitely  bounded  on  all  sides  and  are  not  shown  in  Fig.  5  of  this 
article.  The  areas  of  grouped  residuals  show  the  intercepts  across 
areas  of  like  sign,  but  at  least  two  intercepts  at  an  angle  to  each 
other  are  necessary  to  define  well  the  limits  of  the  area  of  which 
they  are  a  part.  As  the  deflection  stations  are  situated  largely 
in  lines  or  zones  across  the  country  and  not  surrounding  the  areas 
of  like  sign,  it  is  seen  why  the  boundaries  of  relatively  few  areas 
are  well  determined.  In  so  far,  however,  as  the  relations  of  the 
areas  of  positive  and  negative  anomaly  to  positive  and  negative 
deflections  of  the  vertical  are  apparent,  Hayford  and  Bowie  state: 
"The  gravity  anomalies  corroborate  the  evidence  given  by  the 
deflections.  In  no  important  case  are  the  anomalies  and  deflections 
contradictory."2 

It  is  seen  by  inspection  of  the  illustrations  by  Hayford,  and 
also  by  the  discussion  in  Part  II  of  this  article,  that  the  areas  of 

'P.  153,  Part  II. 

2  Hayford  and  Bowie,  1912,  p.  112. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  235 

like  sign  of  deflection  residuals  are  more  sharply  bounded  and 
smaller  in  size  than  the  areas  of  like  sign  of  gravity  anomalies. 
The  latter  occur  commonly  in  areas  so  broad  that  a  vertically  bal- 
anced irregularity  in  the  distribution  of  density  would  have  but 
little  effect.  Yet  the  large  gravity  anomalies  occur  in  the  midst  of 
such  large  areas,  as  shown  on  Fig.  5.  There  are,  furthermore,  few 
sharp  reversals  of  sign  of  the  gravity  anomalies  save  those  at 
different  elevations  in  mountainous  regions  and  these  are  explained 
by  the  presence  of  regional  compensation.  There  are,  on  the 
contrary,  many  sharp  reversals  of  the  deflection  residuals. 

It  is  to  be  concluded,  therefore,  that,  although  some  degree  of 
balancing  of  irregularities  in  the  same  column  no  doubt  exists, 
this  is  not  a  common  or  controlling  explanation  of  the  anomalies 
and  residuals.  They  are  overshadowed  by  a  distribution  which 
points,  on  the  contrary,  to  regional  departures  from  isostasy  by 
regional  excesses  or  defects  in  density. 

In  the  location  of  stations,  the  deflection  observations  are 
arranged  at  relatively  close  intervals  and  in  linear  zones,  owing  to 
the  necessity  of  triangulation.  They  give  the  most  information 
as  to  the  size  of  areas  of  relative  excess  and  defect.  But  two  areas 
of  relative  excess  and  defect  may  both  be  in  absolute  excess  or 
absolute  defect.  The  gravity  stations  are  more  widely  scattered. 
The  local  variations  are  in  consequence  poorly  denned,  but  the 
limits  of  absolute  excess  and  defect  of  mass  are  determined  with 
more  accuracy.  They  appear  to  show  that  areas  as  large  as  1,000 
by  2,000  km.,  620  by  1,240  miles,  may  depart  in  one  direction  from 
isostasy,  but  only  to  a  moderate  amount.  It  is  seen  from  Fig.  5 
that  between  Florida  and  a  line  drawn  from  Lake  Superior  to  the 
Rio  Grande  the  broad  areas  of  less  than  mean  anomaly  are  negative. 
From  this  line  a  great  positive  area  extends  to  the  northwest.  The 
quarter  of  the  United  States  bordering  the  Pacific  Ocean  is,  how- 
ever, another  great  region  of  negative  anomalies.  Upon  these 
broad  regions  of  mean  anomaly  or  less  are  superposed  smaller  and 
better-defined  areas  of  more  than  mean  anomaly,  negative  and  posi- 
tive areas  occurring  in  the  same  broad  region.  These  smaller  areas 
are  inclosed  by  the  0.020  anomaly  contour.  They  commonly 
range  from  300  to  400  km.  across,  200  to  250  miles,  but  the  maxima 


236  JOSEPH  BARRELL 

which  reach  above  o .  040  are  much  smaller.  The  limits  of  regional 
isostasy  appear  then  to  vary  with  the  amount  of  the  load.  Well- 
defined  areas  200  to  250  miles  in  breadth  may  stand  vertically  800 
to  i ,600  feet  on  the  average  from  the  level,  giving  isostatic  equilib- 
rium, and  their  central  portions  reach  still  higher  values.  They 
represent  the  limits  of  regional  isostasy  discussed  in  an  earlier  part. 
But  these  are  superposed  on  broader  areas  which  may  extend  for  a 
thousand  miles  or  more  and  lie  as  much  as  400  to  800  feet  either 
above  or  below  the  level  for  equilibrium.  Stresses  given  by  loads 
of  this  order  are  then  not  restricted  in  area  to  the  limits  set  for 
higher  values. 

The  size  of  the  areas  of  intenser  stress  reveal  the  capacity  to 
which  the  earth  can  carry  mountain  ranges  uncompensated  by 
isostasy.  The  size  of  the  areas  of  weaker  stress  shows  the  capacity 
of  a  considerable  portion  of  a  continent  to  lie  quiescent  while  the 
surface  agencies  carry  forward  their  leveling  work.  This  is  the 
present  state  of  this  particular  continent  after  a  geologic  period  of 
world-wide  notable  vertical  movement  and  adjustment.  It  is  not 
likely,  therefore,  that  these  loads  measure  the  maximum  stress- 
carrying  capacity  of  the  earth.  They  may  be  more  in  the  nature 
of  residual  stresses  which  the  earth  can  hold  through  periods  of 
discharge  of  stress.  East  of  the  Cordillera  there  has  been  but  little 
local  differential  movement  and  these  areas  have  lain  in  crustal 
quiet  for  long  geologic  ages,  being  subject  only  to  broad  and  uni- 
form crustal  warping  of  moderate  amount.  It  is  to  be  presumed, 
therefore,  that  the  strains  which  exist  in  such  regions  by  virtue  of 
the  regional  departures  from  isostasy  are  of  ancient  date  and  well 
within  the  limits  of  crustal  strength. 

It  would  seem  probable  for  such  conditions,  from  the  stand- 
point of  mechanics,  that  the  zone  of  compensation  is  not  sharply 
limited,  with  its  implication  of  marked  lowering  of  rigidity  at  its 
base;  nor  the  distribution  of  compensation  uniform  to  the  base. 
It  seems  more  probable  that  the  abnormalities  of  density  and  the 
resultant  strains  should  fade  out  through  a  considerable  depth 
more  after  the  manner  suggested  by  Chamberlin. 

[To  be  continued] 


VOLUME  XXII  NUMBER  4 


THE 


JOURNAL   OF   GEOLOGY 

MAY-JUNE  1914 


THE  STRENGTH  OF  THE  EARTH'S  CRUST 


JOSEPH  BARRELL 
New  Haven,  Connecticut 


PART  IV.    HETEROGENEITY  AND  RIGIDITY  OF  THE  CRUST  AS 
MEASURED  BY  DEPARTURES  FROM  ISOSTASY 

INTRODUCTION  AND  SUMMARY 289 

VARIABLE  OR  CONSTANT  DEPTH  OF  COMPENSATION       ....  291 
DEPARTURES  FROM  ISOSTASY  SUSTAINED  BY  RIGIDITY  IN  THE  ZONE  OF 

COMPENSATION  .                             296 

INTERPRETATION  OF  DEFLECTION  RESIDUALS  IN  TERMS  OF  MASSES  .  297 

MAXIMUM  LOADS  INDICATED  BY  ANOMALIES .    .  302 

FURTHER  GEODETIC  WORK  NEEDED  FOR  GEOLOGIC  PROBLEMS    ...  313 

INTRODUCTION   AND    SUMMARY 

In  Part  I  were  examined  certain  geologic  tests  of  the  strength  of 
the  crust;  in  Part  II  the  geodetic  evidence  in  regard  to  the  effective 
areal  limits  of  rigidity;  in  Part  III  the  influence  of  variable  vertical 
distribution  of  density.  All  three  lines  of  investigation  converge 
toward  showing  the  rigidity  of  the  outer  crust — the  zone  of  isostatic 
compensation — to  be  such  that  very  considerable  stresses  can  be 
carried  over  areas  whose  radii  range  between  100  and  300  km. 
There  arise  to  be  considered  next  the  following  problems:  first, 
the  variability  in  depth  of  compensation  and  its  influence;  second, 

Vol.  XXII,  No.  4  289 


290  JOSEPH  BARRELL 

whether  the  stresses  represented  by  the  incompleteness  of  isostasy 
are  carried  by  the  rigidity  of  the  outer  crust,  or  are  transferred 
in  some  measure  to  the  deeper  body  of  the  earth;  third,  the  magni- 
tudes of  the  stresses,  measured  in  terms  of  loads,  which  are  indi- 
cated by  the  gravity  anomalies  and  deflection  residuals. 

It  is  found  in  answer  that  under  the  hypothesis  which  forms 
the  basis  of  Hayford's  work,  that  of  uniform  compensation,  com- 
plete at  a  given  depth,  there  are  indications,  given  by  comparing 
different  areas,  of  a  great  range  in  the  depth  of  the  bottom.  Under 
an  assumption  which  is  probably  nearer  to  nature — that  is,  the 
hypothesis  of  a  variable  and  gradually  disappearing  compensa- 
tion— there  is  room  for  even  a  greater  heterogeneity  of  the  crust 
and  a  greater  variability  in  the  depth  reached  by  the  zone  of 
compensation.  But,  on  the  other  hand,  it  is  concluded  that  the 
zone  of  compensation,  as  an  outer  rigid  crust  separated  from  the 
rigid  inner  earth  by  an  intervening  zone  of  lowered  rigidity,  is  a 
reality  in  earth  structure.  The  stresses  due  to  the  heterogeneities 
of  density  and  relief  within  and  upon  this  crust  appear  to  be  borne 
by  the  crust,  not  by  the  inner  earth.  Under  the  third  subject  it 
appears,  upon  review  of  the  evidence  given  by  the  deflection 
residuals,  that  these  may  be  interpreted  so  as  to  show  departures 
from  equilibrium  comparable  to  the  results  given  by  the  gravity 
anomalies,  instead  of  the  250  feet  which  Hayfofd  thought  to  exist. 
The  two  independent  lines  of  geodetic  investigation  are  thus  seen 
to  agree  and  it  may  be  concluded  with  some  confidence  that  the 
individual  isostatic  regions  of  the  United  States  are  on  the  average 
between  600  and  900  feet  out  of  equilibrium.  Evidence  from  other 
parts  of  the  world  appears  to  show,  furthermore,  that  a  number  of 
regions  exhibit  greater  departures  from  isostasy  than  those  observed 
within  the  United  States.  The  strain  imposed  on  the  crust  by  the 
Niger  Delta,  though  large,  is  apparently  not  as  large  as  some  made 
known  by  geodetic  measurements. 

Thus  from  various  directions  of  attack  the  crust  is  shown  to  be 
an  earth  shell  of  high  rigidity  and  consequently  high  elasticity. 
Geodetic  evidence  justifies  the  view,  brought  forward  by  geologic 
evidence,  that  the  delta  of  the  Niger  is  to  be  looked  upon  as  sup- 
ported by  the  strength  of  the  crust. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  291 

VARIABLE    OR   CONSTANT   DEPTH   OF   COMPENSATION 

The  Cordilleran  region  awoke  to  an  era  of  great  orogenic  and 
igneous  activity  near  the  beginning  of  the  Tertiary,  and,  especially 
in  the  Neocene,  has  become  broadly  elevated  into  one  of  the  great 
plateau  regions  of  the  world.  Large  areas  like  the  Colorado 
plateaus,  which  since  the  beginning  of  the  Paleozoic  had  rested  near 
sea-level,  at  times  beneath  and  again  slightly  above,  have  been  lifted 
many  thousands  of  feet.  Block-faulted  structures  indicate  the 
dominance  of  vertical  forces  rather  than  surficial  compression  as 
the  cause  of  these  movements.  The  uplift  has  not  been  of  the 
nature  of  a  broad  even  upwarp,  and  adjacent  regions  show  great 
contrasts  in  elevation.  These  different  surface  results  of  the 
interior  forces  suggest  differences  in  elevatory  forces  at  compara- 
tively shallow  depths.  The  region  is  known  to  be  in  a  fair  degree 
of  isostatic  equilibrium  notwithstanding  the  high  relief.  Davis  has 
shown  why  these  movements  cannot  be  regarded  as  differential 
sinkings  toward  the  center  of  the  earth.1  These  features  suggest, 
then,  subcrustal  decreases  in  density  during  the  Tertiary  as  a  cause 
of  the  broad  movements  of  elevation. 

The  rising  of  great  bodies  of  magma  to  high  levels  in  the  zone  of 
isostatic  compensation,  their  irregular  distribution,  the  great 
quantities  of  heat  and  gases  which  would  invade  the  roofs  are 
suggested  by  the  observed  evidences  of  regional  igneous  activity  at 
the  surface  as  the  probable  causes  of  the  changes  in  density  and 
regional  vertical  movements.  A  consequence  of  such  a  cause 
would  be  a  lessened  strength  of  the  crust  to  resist  strain,  a  lessened 
depth  to  the  zone  of  isostatic  compensation,  and  a  decreased  size  of 
the  unit  areas  departing  from  equilibrium. 

The  history  of  the  Cenozoic  in  the  Cordillera  has  repeated  the 
history  of  other  regions  at  other  times,  either  in  the  Archean 
igneous  activity  or  later.  The  slow  conduction  of  this  excess  heat 
from  the  outer  crust,  the  solidification  of  the  reservoirs  of  magma, 
would,  in  the  course  of  ages,  bring  about  a  new  rigidity.  Upon 
disturbances  of  the  equilibrium  by  erosion  or  compressive  forces 
there  would  be  found  a  new  and  greater  depth  to  the  zone  of 
compensation. 

l"  Bearing  of  Physiography  upon  Suess's  Theories  Abstract,"  Intern.  Geog. 
Cong.,  8th  Report,  1905,  p.  164;  Amer.  Jour.  Science  (4),  XIX  (1905),  265-73. 


292  JOSEPH  BARRELL 

Where  ages  of  uplift  and  erosion  have  followed  periods  of 
igneous  activity  there  are  revealed  great  bodies  of  intrusive  rock 
varying  in  density  from  granites  at  2.65  to  gabbros  at  3 .  o.  These 
great  batholiths  are  of  irregular  distribution  in  the  crust,  both 
vertically  and  horizontally.  Their  abundance  increases  downward 
so  far  as  erosion  has  revealed  the  evidence.  The  outer  crust  of  the 
earth  has  become  vertically  and  areally  heterogeneous  by  such 
means  and  should  cause  variations  and  irregularities  to  an  appre- 
ciable degree  in  the  distribution  of  isostatic  compensation,  as  noted 
under  the  topic  of  the  influence  of  variable  rate  of  compensation 
upon  gravity  anomalies.  Here  we  note  in  addition  the  decreased 
depth  of  compensation  and  decreased  rigidity  at  the  time  of 
intrusion. 

Hayford  notes  that  the  stations  classified  into  geographic  groups 
show  as  a  rule  as  great  contradictions  in  depths  of  compensation 
between  adjacent  groups  as  in  those  which  are  far  apart.  This 
variation  between  adjacent  groups  is  taken  by  him  as  weakening 
the  evidence  that  there  is  any  real  variation  in  the  depth  of 
compensation  over  the  whole  area  investigated.1  For  the  reasons 
outlined  previously,  the  present  writer,  influenced  by  the  geologic 
inferences,  does  not  view  such  irregularity  of  distribution  as  proof 
that  the  evidence  is  weak  and  conflicting.  The  strength  of  the 
evidence  must  be  judged  rather  by  the  nature  of  the  residuals. 
Hayford  points  out  that  the  depth  of  compensation  in  the  West 
seems  on  the  whole  to  be  somewhat  less  than  in  other  parts  of  the 
United  States,  though  he  does  not  regard  it  as  safe  to  assert  that  it 
does  exist.  On  dividing  the  whole  area  into  four  sections,  the 
minimum  sum  of  the  squares  of  the  residuals  indicates  depths  as 
follows  as  best  satisfying  the  hypothesis  of  uniform  distribution  of 
compensation: 

From  all  residuals  of  the  central  group,  174  km. 
From  all  residuals  of  the  northeastern  group,  187  km. 
From  all  residuals  of  the  southeastern  group,  indeterminate. 
From  all  residuals  of  the  western  group,  107  km.2 

In  the  1909  paper  Hayford  gives  a  tabulation  of  the  residuals  for 
fourteen  geographic  groups.     The  results  for  the  United  States  as  a 
1 1909,  pp.  58, 59.  2 1906,  pp.  142-46. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST 


293 


whole  and  for  the  groups  showing  the  shallowest,  deepest,  and  the 
most  irregular  compensation  are  quoted  below.1  That  solution  is 
regarded  as  nearest  the  truth  which  gives  the  smallest  mean  value 
of  the  squares  of  the  residuals. 


TABLE  XIX 

PROBABLE  DEPTHS  OF  COMPENSATION 


MEAN  VALUES  OF  THE  SQUARES  OF  THE 

RESIDUALS  IN  VARIOUS  GROUPS 

No.  OF 

MOST 

GROUP 

RESIDU- 
ALS 

Solution 

Solution 

Solution 

Solution 

Solution 

PROB- 
ABLE 

8 

B 

E 

H 

G 

A 

DEPTH 

Infinite 

162.2 

120.  Q 

"3-7 

o.o 

Depth 

Km. 

Km. 

Km. 

Km. 

Km. 

United  States  (all  observa- 

tions) 

772 

146.  50 

14  cx 

17.71 

17    7tr 

2<    77 

122.2 

Group  12  (parts  of  Minn., 

/  oo 

AiT  '  VO 

*  O     1  O 

o    /  o 

Oil 

N.Dak.,  S.Dak.,  Neb., 

Kan.)  

36 

196.57 

7.OO 

7-47 

7.59 

II  .46 

305 

Group  5   (Mich.,  Minn., 

Wis.) 

52 

34-97 

23.6O 

23.64 

23.67 

27-53 

152 

Group  8  (parts  of  Utah, 

Nev.,  CaL).  .  . 

42 

128.97 

22.  27 

18.79 

18.25 

35.78 

66 

United   States   (residuals 

multiplied  by  1.327  to 

compare  with  Group  8) 

194.40 

18.65 

18.23 

18.25 

34-21 

It  is  seen  that  the  mean  value  of  the  squares  of  the  residuals  in 
group  12  with  most  probable  depth  of  305  km.  is  considerably  less 
than  for  the  United  States  as  a  whole,  in  part  no  doubt  owing  to  the 
moderate  relief,  yet  the  differences  between  the  residuals  in  group 
1 2  for  the  different  solutions  is  much  more  pronounced  than  for  the 
United  States  as  a  whole.  The  number  of  stations,  36,  is  large 
enough  so  that  this  can  hardly  be  regarded  as  accidental.  On  the 
contrary,  it  would  appear  that  for  the  whole  United  States  the 
group  differences  are  sufficient  to  mask  in  part  the  accuracy  of  the 
mean  result  of  122  km.  and  that  the  depth  of  compensation  within 
certain  groups  is  more  reliable  than  for  the  United  States  as  a  whole. 

In  group  8  with  most  probable  depth  of  66  km.  the  mean  value 
of  the  squares  of  the  residuals  is  nearly  50  per  cent  higher  than  for 

1 1909,  pp.  55-58. 


294  JOSEPH  BARRELL 

the  United  States  as  a  whole,  a  value  which  may  be  ascribed  to  the 
mountainous  relief  and  the  support  of  individual  mountains  and 
ranges  by  the  rigidity  of  the  crust.  Nevertheless  the  residuals  for 
the  several  solutions  fall  into  a  somewhat  regular  system,  and  solu- 
tions E,  H,  and  G  are  more  sharply  differentiated  from  the  most 
probable  one  than  for  the  whole  United  States.  They  may  be 
compared  better  with  the  latter  if  the  residuals  for  the  whole 
country  are  multiplied  by  i .  327  as  a  factor  in  order  to  give  the  same 
numerical  value  under  solution  G.  This  is  done  at  the  bottom  of 
the  table.  It  would  appear  from  these  figures  as  though  the  argu- 
ments previously  given  from  geologic  analysis  receive  considerable 
support  from  the  geodetic  results  and  point  to  a  much  shallower 
depth  for  isostatic  compensation  in  the  Great  Basin  than  over  cer- 
tain other  portions  of  the  United  States.  Furthermore,  in  the 
examination  of  the  question  of  local  versus  regional  compensation, 
it  was  only  the  forty  mountain  stations  classified  into  two  groups 
according  to  elevation  which  gave  any  suggestion  that  regional 
compensation  to  a  radial  distance  of  166.7  km.  was  not  about  as 
probable  as  more  local  compensation.  In  these  two  lines  of  geodetic 
evidence  as  to  limited  depth  and  breadth  of  compensation  there  are 
suggestions  therefore  which  support  the  geologic  inference  that  the 
crust  of  the  Cordilleran  region  may  be  weaker  than  over  the  United 
States  as  a  whole.  On  the  other  hand,  the  warping  or  faulting- 
down  of  ancient  continental  areas  into  marginal  sea-bottoms 
implies  an  increasing  density  of  the  subcrust  and  therefore  possibly 
an  increasing  rigidity  and  strength  under  such  areas.  Such  a 
contrast  between  the  Atlantic  Ocean  bottom  and  the  Great  Basin 
would  correspond  to  the  great  strength  of  crust  necessary  to  sustain 
the  delta  of  the  Niger  as  compared  with  the  moderate  rigidity  found 
by  Gilbert  for  the  crust  beneath  extinct  Lake  Bonneville,  located 
within  the  limits  of  group  8. 

The  regions  of  shallower  compensation  in  the  United  States  are 
on  the  whole  marked  probably  by  a  higher  temperature  gradient, 
the  regions  of  deep  compensation  by  a  lower.  This  is  illustrated  by 
the  very  high  gradient  of  the  Comstock  mine  in  Nevada  and  the 
very  low  gradient  which  is  found  in  the  Lake  Superior  copper  mines. 
The  temperature  gradient  may  measure  the  depth  to  a  zone  of  low 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  295 

rigidity,  determined  by  a  certain  relation  of  temperature  and 
pressure. 

Within  an  overlying  zone  of  high  rigidity,  even  where  it  is  of 
uniform  depth,  the  geodetic  measurements  of  the  depth  of  com- 
pensation may  not,  however,  show  uniformity.  If  the  density  is 
unequally  distributed,  the  compensation  of  a  region  may  be  nearly 
completed  at  some  depth  above  the  base  of  the  rigid  zone,  the  lower 
part  consisting  of  rock  of  mean  density  and  therefore  not  possessing 
influence.  A  region  of  deep  and  marked  rigidity,  if  characterized 
by  notable  irregularities  in  the  distribution  of  either  density  or 
relief,  would  show  large  residuals.  A  region  characterized  by  more 
uniform  distribution  of  density  and  gentle  relief  would  show  lower 
residuals  even  with  the  same  rigidity.  A  region  with  deep  com- 
pensation would  show  within  the  limits  of  the  group  lower  residuals 
for  the  same  degree  of  uniform  compensation,  than  where  compen- 
sation was  at  lesser  depths,  since  the  attracting  masses  are  spread 
over  a  greater  distance. 

As  applications  and  tests  of  these  principles,  it  is  to  be  noted 
that  group  5,  embracing  the  Lake  Superior  region  with  its  low- 
temperature  gradients,  has  the  highest  residuals  of  any  group  in  the 
United  States.  Further,  the  mean  values  of  the  least  squares  for 
the  different  solutions  show  less  differentiation  than  in  any  other 
group.  These  facts  suggest  irregular  distribution  of  density,  high 
rigidity,  and  the  zone  of  rigidity  may  extend  below  the  most 
probable  depth,  152  km.,  indicated  for  the  limits  of  compensation. 
The  topographic  deflections  are  only  58  per  cent  compensated. 
The  contiguous  group  to  the  southwest,  No.  12,  shows  the  lowest 
residuals  of  any  group,  the  separate  solutions  are  sharply  differen- 
tiated and  the  depth  is  the  greatest  in  the  United  States.  On  the 
side  of  this  area,  the  gravity  anomaly  at  St.  Paul,  0.059  dyne  per 
gram,  is,  next  to  Seattle,  the  largest  found  thus  far  in  the  United 
States.  It  may  be  concluded,  then,  that  in  this  part  of  the  con- 
tinent, undisturbed  by  igneous  activity  or  mountain-building  since 
the  pre-Cambrian,  the  depth  of  the  zone  of  rigidity  appears  to  be 
very  great.  The  irregularities  in  residuals  in  group  5  may  date 
from  the  Keweenawan  period,  when  enormous  masses  of  basic  and 
therefore  heavy  magmas  were  intruded  and  extruded  in  the  Lake 


296  JOSEPH  BARRELL 

Superior  region.  If  such  be  the  case  it  shows  the  long  endurance  of 
strains  borne  by  this  part  of  the  earth.  In  the  almost  universal 
epeirogenic  movements  which  marked  the  close  of  the  Tertiary  and 
opening  of  the  Pleistocene,  the  Lake  Superior  basin  showed  notable 
down  warping,  its  bottom  being  now  beneath  the  level  of  the  sea. 
It  formed  a  trough  which  directed  the  flow  of  glacial  ice.  The 
latter  must  have  scoured  it  clean  but  can  hardly  be  ascribed  as  the 
cause  of  the  existence  of  the  basin.  The  crust  movements  have 
doubtless  been  in  the  direction  of  relief  of  stress,  but  the  relief  has 
been  but  partial;  geodetic  investigation  reveals  that  the  age-long 
load  is  yet  borne. 

DEPARTURES  FROM  ISOSTASY   SUSTAINED   BY  RIGIDITY  IN   THE  ZONE 

OF   COMPENSATION 

It  was  concluded  under  the  last  topic  that  the  rigidity  over 
certain  parts  of  the  earth  probably  carries  the  zone  of  possible  com- 
pensation as  deep  as  300  km.  even  under  the  assumption  of  uniform 
rate,  an  assumption  which  tends  to  minimize  the  depth;  whereas 
in  other  regions  under  that  hypothesis  it  is  less  than  100  km.  in 
depth.  This  raises  the  question  whether  the  regional  departures 
from  isostasy  are  carried  as  strains  within  the  zone  of  compensation 
or  are  transferred  in  part  to  the  deeper  body  of  the  earth.  There 
are  reasons  for  believing  that  the  former  is  the  case,  pointing  by 
inference  to  a  zone  of  markedly  diminished  rigidity  between  the 
rigid  lithosphere  and  still  more  rigid  centrosphere. 

The  geodetic  evidence  consists  in  the  large  values  of  the  squares 
of  the  residuals  for  solution  B,  the  solution  which  postulates  extreme 
rigidity  and  compensation  at  infinite  depth.  For  the  whole  United 
States,  as  shown  in  the  Table  XIX,  p.  293,  the  mean  value  of  the 
squares  of  the  residuals  for  solution  B  is  10.7  times  the  value  for 
solution  H.  But  for  group  12,  that  for  which  the  most  probable 
depth  of  compensation  is  305  km.,  the  distinction  is  still  greater; 
solution  B  showing  a  mean-square  residual  28  times  greater  than 
for  solution  E.  Dividing  in  this  way  the  value  for  solution  B  by  the 
value  for  the  most  probable  solution,  and  taking  the  mean  for  all 
those  groups  which  indicate  a  depth  of  compensation  greater  than 
the  average  for  the  United  States,  it  is  found  that  the  ratio  is  twice 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  297 

as  great  for  the  groups  with  deep  compensation  as  for  the  United 
States  as  a  whole.  That  is,  the  groups  with  deep  compensation, 
instead  of  showing  a  leaning  toward  solution  B  show  on  the  con- 
trary more  definitely  that  it  is  not  true.  The  hypothesis  of  uniform 
compensation  complete  at  a  certain  depth  appears  to  be  more  nearly 
true  for  regions  with  deep  compensation  than  for  shallow  com- 
pensation. This  does  not  mean,  however,  a  lesser  rigidity  of  the 
crust  for  the  regions  with  deep  compensation,  their  high  capacity 
to  carry  strain  being  shown  by  the  large  gravity  anomalies  which 
are  found  in  places  within  them. 

There  seems  to  be  no  evidence,  however,  that  the  zone  of 
diminished  rigidity  is  sharply  bounded  or  is  marked  by  real  liquidity. 
It  is  doubtless  due  to  the  gradual  rise  of  temperature  with  depth, 
overcoming  within  a  certain  zone  the  influence  of  the  increasing 
pressure.  Seismologic  and  tidal  evidences  show,  furthermore,  that 
under  stresses  of  relatively  brief  duration  the  earth  acts  as  a  unit 
and  as  an  elastic  rigid  body.  The  physical  condition  of  the  zone  of 
low  rigidity  may  approach  that  of  a  highly  viscous  fluid,  the  time 
element  thus  entering  within  these  limits  as  a  fundamental  factor. 
This  zone  is  incapable  of  bearing  pronounced  strains  for  long 
periods  in  the  manner  of  the  zone  above.  In  geologic  operations 
it  thus  serves  to  separate  the  mode  of  expression  of  forces  gen- 
erated below  from  those  originating  above  this  level.  The  former 
give  rise  to  the  great  compressive  movements  in  the  outer  zone, 
the  latter  to  the  vertical  movements  not  determined  by  tangential 
compression. 

INTERPRETATION   OF   DEFLECTION   RESIDUALS   IN   TERMS   OF  MASSES 

On  p.  59,  paper  of  1909,  Hayford  shows  that  the  actual  deflec- 
tions of  the  vertical  average  only  one-tenth  of  what  they  would  be 
if  the  continent  and  the  portions  of  the  ocean  basins  which  were 
included  in  the  calculations  were  both  underlain  by  matter  of  the 
same  density  and  the  relief  sustained  wholly  by  the  rigidity  of  the 
crust.  The  effect  of  the  topography  calculated  on  this  assumption 
— -that  the  density  is  uniform  and  the  larger  as  well  as  the  smaller 
features  are  sustained  by  rigidity — gives  what  is  known  as  the 
topographic  deflections.  These,  as  stated  above,  average  ten  times 


298 


JOSEPH  BARRELL 


the  value  of  the  actually  observed  deflections.  The  surface  may  be 
regarded,  therefore,  as  nine-tenths  compensated  by  variations  of 
density.  The  details  for  the  five  more  significant  groups  are 
given  below:1 

TABLE  XX 


i 

2 

3 

4 

S 

6 

7 

8 

d 

"o  cji 

•M£$ 

°1 

•s^S 

Sggffi 

No. 

Area  of  Group 

1 

C/2   § 

ij 

•&  *Si    O    4> 

Jg-3*S 
op  g  S 

can  of  Topograp 
Deflections  wi 
out  Regard  to  S 

can  Residual  of 
lution  H  withe 
Regard  to  Sign 

sill 

rcentage  of  Co 
pleteness  of  ] 
static  Compen 
tion  for  Solution 

* 

dn 

% 

S 

> 

£ 

12.  .  .  . 

Parts    of    Minn.,    N.Dak., 

S.Dak.,  Neb.,  Kan  

36 

305 

8.23 

2.17 

o.  26 

74 

8  

Parts  of  Utah,  Nev.,  Cal  

42 

66 

32-23 

3-57 

.  II 

89 

IO     .  .  . 

Cal.,  southern  part  

C7 

126 

65  44 

3  91 

06 

O4. 

9  

Cal.,  northern  part  

60 

176 

60.50 

2-93 

•  05 

95 

14  

Northern  Cal.,  western  Ore., 

and  Wash 

37 

84. 

53  68 

337 

06 

QA 

Whole  United  States 

733 

122 

3O    37 

2    01 

O    IO 

no 

Group  12  gives  the  greatest  depth  for  uniform  compensation. 
By  using  the  residual  for  Solution  E,  2.09,  the  percentage  of 
completeness  of  compensation  would  have  been  75,  a  trifle  more 
than  for  Solution  H,  but  still  next  to  the  least  perfect  in  the  United 
States. 

Group  8,  the  Great  Basin  region,  has  the  lowest  depth  of  com- 
pensation but  shows  about  the  average  approximation  to  isostatic 
equilibrium. 

Groups  10,  9,  14  comprise  the  Pacific  Coast  Ranges.  They  give 
the  highest  topographic  deflections  of  the  United  States,  doubtless 
on  account  of  the  great  relief  of  the  ocean  basin  and  continental 
border,  but  the  actually  observed  deflections  do  not  differ  greatly 
from  group  8  or  the  mean  for  the  whole  United  States.  The  result 
is  that  in  this  mountain  region  bordering  the  continent  the  degree 

1  Taken  from  pp.  56,  58,  69,  and  illustration  No.  2,  Hay  ford,  1909. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  299 

of  completeness  of  compensation  is  the  highest  in  the  United  States. 

On  the  basis  of  the  figures  for  the  whole  United  States  Hayford 
writes:  "The  average  elevation  above  mean  sea-level  being  about 
2,500  feet,  this  average  departure  of  less  than  one-tenth  from  com- 
plete compensation  corresponds  to  excesses  or  deficiencies  of  mass 
represented  by  a  stratum  only  250  feet  (76  meters)  thick  on  an 
average."1  It  is  this  last  statement,  interpreting  the  deflection  in 
terms  of  mass,  which  has  meaning  to  the  geologist.  It  has  been 
widely  quoted  as  perhaps  the  chief  geologic  result  of  the  work  and 
yet  the  writer  believes  that  it  is  without  basis.  By  an  oversight  of 
the  author  he  misinterprets  his  results.  If  the  present  writer  is 
correct  in  making  this  statement  it  should  not  be  taken,  however,  as 
a  criticism  of  the  mathematical  portion  of  the  work. 

The  sea-level  is  from  the  standpoint  of  the  problem  of  isostatic 
compensation  but  little  more  than  a  datum  surface.  Imagine  the 
ocean  water  to  be  converted  into  rock  of  density  2 . 7  of  the  same 
mass  as  the  water  and  resting  on  the  present  ocean  bottom.  Every 
thousand  feet  of  water  would  be  replaced  by  380  ft.  of  rock.  Then 
the  sea-level  surface  after  this  transmutation  is  seen  to  lose  all  real 
significance.2  To  show  the  fallacy  of  taking  this  level  as  a  basis  for 
interpreting  the  departures  from  compensation  in  terms  of  thick- 
nesses, let  attention  be  given  to  groups  i,  2,  3,  4,  6,  n,3  which  cover 
the  United  States  east  of  the  Mississippi  River.  The  average 
departure  of  these  from  compensation  is  o .  1 1 ,  which  on  the  basis  of 
Hayford's  statement  means  that  the  surface  on  the  average  departs 
but  275  ft.  from  the  level  which  would  give  complete  isostatic 
equilibrium  on  the  hypothesis  of  uniform  distribution  of  compensa- 
tion to  a  depth  of  122  km.  If,  however,  this  eastern  third  of  the 
United  States  be  regarded  by  itself,  its  average  elevation  may  be 
assumed  as  1,000  ft.  (it  is  probably  less).  By  the  same  reasoning  as 
Hayford  applied  to  the  whole  United  States,  n  per  cent  of  this  is 
no  ft.  Therefore  although  the  average  deflections  are  slightly 

1 1909,  p.  59. 

2  More  accurately,  the  equivalent  rock  should  be  imagined  as  suspended  at  the 
mean  depth  of  the  water,  but  the  effect  of  the  difference  in  level  is  negligible  upon  the 
topographic  deflection. 

3  1909,  p.  59. 


300  JOSEPH  BARRELL 

greater  than  for  the  United  States  as  a  whole,  it  would  be  concluded 
that  for  the  region  east  of  the  Mississippi  the  departure  from  the 
levels  giving  complete  compensation  averages  not  more  than  no 
ft.  instead  of  the  275  ft.  previously  stated. 

Or,  again,  imagine  a  rise  of  ocean-level  so  that  the  average  eleva- 
tion of  this  part  of  the  continent  is  reduced  to  100  ft.  without 
changing  the  detail  of  the  topography.  The  deflections  would  suffer 
'only  small  alterations  due  to  the  added  mass  of  water.  Although 
the  crust  remained  without  change,  the  same  reasoning  would  then 
lead  to  the  conclusion  that  the  topography  departed  on  the  average 
but  n  ft.  from  the  levels  which  would  give  complete  compensation. 

In  computing  the  influence  of  the  topographic  irregularities  and 
their  compensation  upon  the  deflection  of  the  vertical,  all  the 
topography  was  taken  into  account  up  to  a  radial  distance  of 
4,125  km.  from  each  station.  This  radius  is  approximately  the 
length  of  37°  of  latitude.  It  embraces  the  Pacific  Ocean  out  to  the 
Hawaiian  Islands  and  to  ten  degrees  south  of  the  equator,  and  the 
Atlantic  Ocean  out  to  the  Azores.  The  relief  within  this  region 
ranges  from  —8,340  m.  north  of  Porto  Rico  to  +6,220  m.  in  Mount 
McKinley,  +6,247  m  Chimborazo;  a  total  differential  relief  of 
about  14,590  m.  About  one-half  of  the  topography  surrounding 
the  coast  stations  consists  of  ocean  bed.  Even  for  the  stations  in 
Minnesota,  farthest  removed  from  the  sea,  about  one-third  of  the 
surrounding  topography  within  the  limits  is  deep  ocean,  but  lying 
at  a  greater  distance  and  carrying  lesser  influence.  The  average 
depth  of  the  oceans  influencing  the  deflection  of  the  station  at  mean 
distance  inland  may  be  assumed  for  purposes  of  illustration  to  be 
about  5,000  meters.  This  depth  of  water  is  equivalent  in  mass  to 
1,900  m.  of  rock  of  density  2 . 67,  leaving  an  effective  ocean  depth  of 
3,100  m.  Add  the  mean  continental  elevation  of  760  m.  to  this, 
and  3,800  to  3,900  m.  represents  about  the  effective  mean  relief 
between  continent  and  ocean.  On  coast  stations  this  differential 
relief  has  greatest  influence.  For  inland  stations  the  several 
portions  of  the  continent  have  proportionately  more  effect.  For 
the  United  States  as  a  whole  it  is  this  relief  of  between  3,500  and 
4,000  m.  between  continent  and  ocean,  more  than  the  relief  between 
the  major  features  of  the  continent,  which  is  nine-tenths  compen- 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  301 

sated  by  the  corresponding  variations  in  crustal  density,  not  the 
760  m.  which  is  the  average  elevation  of  the  United  States  above 
sea-level. 

It  is  the  belt  of  Pacific  coast  stations  which  measures  more 
closely  than  other  groups  the  degree  of  compensation  accompany- 
ing the  continental  relief  above  the  ocean  bottoms.  These  stations 
lie  in  groups  10,  9,  and  14,  for  which  the  mean  residuals  are  but 
0.06,  0.05,  and  0.06  of  the  mean  topographic  deflections  respec- 
tively. These  residual  deflections  indicate  that  for  this  coastal  zone 
the  departures  from  complete  compensation  amount  to  but  5  or 
6  per  cent.  If  the  mean  effective  relief  which  controls  this  be 
assumed  as  4,000  m.,  then  the  mean  departure  from  equilibrium  is 
represented  by  a  mass  200  to  240  m.  thick,  approximately  between 
650  and  800  ft.  On  the  other  hand,  groups  5  and  12  are  those 
farthest  removed  from  the  ocean  basins  and  their  deflections  are 
controlled  most  largely  by  the  internal  continental  relations.  For 
them  the  departures  from  complete  isostatic  compensation  as 
measured  by  the  ratio  of  the  mean  residuals  to  the  computed 
topographic  deflections  amount  to  42  and  26  per  cent.  The  mass 
to  which  this  is  equivalent  may  be  no  greater  than  the  5  per  cent 
departure  on  the  Pacific  coast.  These  estimates  fall  into  the  same 
order  of  magnitude  as  that  of  the  masses  represented  by  the 
gravity  anomalies. 

This  reconnaissance  of  the  problem  is  sufficient  for  present  pur- 
poses. It  is  readily  seen  that  even  greater  difficulties  stand  in  the 
way  of  a  precise  statement  regarding  the  equivalence  of  mass 
corresponding  to  deflections  of  the  vertical  than  arose  in  the  inter- 
pretation of  the  gravity  anomalies.  The  residual  for  each  observed 
deflection  is  the  sum  of  the  influences  of  all  the  excesses  and  defi- 
ciencies of  mass  as  compared  to  solution  H  on  all  sides  of  a  station. 
The  effect  of  each  unit  varies  inversely  with  the  square  of  the 
distance  and  directly  with  the  sine  of  the  angle  which  the  line  of 
force  makes  with  the  horizontal  passing  through  each  station.  A 
combination  of  the  data  from  the  measurements  of  the  intensity  of 
gravity  with  those  of  the  deflections  of  the  vertical  would  apparently 
be  necessary  to  state  for  each  region  the  equivalence  in  terms  of 
mass  which  is  implied  by  the  residual  at  each  station. 


302  JOSEPH  BARRELL 

MAXIMUM   LOADS   INDICATED   BY   ANOMALIES 

Hayford  and  Bowie  consider  that  0.0030  dyne  of  anomaly  may 
be  regarded  as  equivalent  to  100  ft.  of  rock  possessing  a  density  of 
2.67.  From  the  previous  considerations  it  would  seem  that  this  is 
probably  too  high  for  a  mean  figure,  but  may  apply  to  certain  areas, 
especially  those  with  extremely  broad  boundaries.  In  other  regions 
0.003  may  be  far  too  high,  since  it  is  shown  under  the  topic  "  Vari- 
able or  Constant  Depth  of  Compensation"  that  in  certain  parts  of 
the  United  States  the  depth  of  the  zone  of  compensation  probably 
goes  notably  deeper  than  in  other  parts  and  the  density  may  be 
distributed  either  nearly  uniformly  or  with  considerable  irregularity. 
The  greatest  depth  of  compensation  indicated  for  any  region  is  305 
km.  A  unit  thickness  of  mass  uniformly  distributed  to  this  depth 
and  to  a  radius  of  166.7  km.  would  give  but  0.0014  of  anomaly 
instead  of  o .  0024  as  given  by  a  depth  of  1 14  km.,  or  o .  0030  as  taken 
by  Hayford  and  Bowie.  For  general  use  0.0024  dyne  is  perhaps 
the  best  value,  corresponding  to  a  uniform  distribution  of  a  unit 
excess  or  defect  of  mass  to  a  depth  of  114  km.  and  to  a  radial  dis- 
tance of  1 66. 7  km.  For  the  mean  anomaly  of  0.018  this  would 
give  750  ft.  of  elevation  as  the  mean  departure  of  the  surface  of  the 
United  States  above  or  below  the  position  giving  isostatic  equi- 
librium, instead  of  600,  or  more  exactly,  630  ft.  as  taken  by  Bowie. 
The  largest  known  anomaly  in  the  United  States  is  at  Seattle, 
—0.093.  This  corresponds  to  a  defect  in  mass  equivalent  to  a 
stratum  4,000  ft.  thick  if  the  divisor  is  0.0024,  a  stratum  3,200 
ft.  thick  if  the  divisor  is  0.0030.  At  Olympia,  but  50  miles  or  80 
km.  distant,  the  anomaly  is  +0.033,  corresponding  to  excesses  of 
mass  of  1,375  or  1,100  ft.,  according  to  the  divisor.  The  difference 
of  regional  load  between  Olympia  and  Seattle  becomes  5,375  or 
4,300  ft. 

But  these  relations  of  unit  thickness  of  mass  to  the  gravity 
anomaly  are  based  on  the  assumption  that  the  excess  or  deficiency 
of  mass  extends  to  as  great  a  radial  distance  as  166.7  km.  radius. 
This  minimizes  the  thicknesses  or  densities  needed  to  account  for 
the  anomalies  above  what  would  be  required  for  a  more  local 
concentration  of  mass.  But  an  inspection  of  the  distribution  of 
gravity  and  deflection  residuals  shows  that  in  many  cases  the  masses 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  303 

producing  the  greater  disturbances  have  much  smaller  size.  This 
is  especially  striking  in  the  case  of  the  largest  negative  anomaly 
in  the  United  States,  that  at  Seattle,  only  50  miles  from  the  large 
positive  anomaly  at  Olympia.  The  latter  is  surrounded  on  all 
sides  by  negative  anomalies  as  follows: 

DISTANCES  FROM  OLYMPIA,  WASHINGTON 

Astoria,  Ore 76  miles  S.W.        — .  013  dyne  anomaly 

Heppner,0re 195     "      S.E.         -.027     " 

Skyhomish,  Wash..   84     "      N.E.        -.028     " 
Seattle,  Wash 50    "      N.N.E.  -.093     " 

The  excess  of  mass  which  exists  in  the  vicinity  of  Olympia,  above 
that  required  for  compensation  under  solution  G,  must  therefore 
be  much  less  than  166.7  km.  (102.5  miles)  in  radius.  The  same 
is  doubtless  true  of  that  excessive  deficiency  which  exists  at  Seattle, 
since  the  anomaly  sinks  to  less  than  one-third  the  value  at  Sky- 
homish only  45  miles  east,  and  changes  to  a  large  positive  anomaly 
at  Olympia,  50  miles  south-southwest. 

The  large  positive  mass  in  the  vicinity  of  Olympia  must  dimin- 
ish appreciably  the  effect  of  the  still  larger  negative  mass  in  the 
vicinity  of  Seattle.  The  latter  with  the  other  surrounding  negative 
masses  must  diminish  still  more  the  anomaly  due  to  the  positive 
mass  at  Olympia.  Furthermore,  it  is  highly  improbable  that  the 
observations  at  Seattle  should  happen  to  be  made  at  the  point  of 
really  maximum  anomaly.  Let  the  very  moderate  assumption  be 
made  then  that  the  abnormal  Seattle  mass  as  a  unit  by  itself 
would  give  a  maximum  anomaly  of  —  o.ioo  dyne.  It  would 
doubtless  give  more.  Let  limiting  assumptions  be  made  as  to  the 
dimensions  and  density  of  this  mass  such  that  the  actual  volume 
and  density  are  quite  probably  embraced  somewhere  within  these 
limits.  Tables  XXI1  and  XXII  show  the  results  of  such  assump- 

1  Table  XXI  is  readily  derived  from  Table  X,  Part  III.  Take,  for  example, 
the  cylinder  of  radius  1,280  meters,  depth  of  1,000  feet,  and  density  o.  267.  Multiply 
its  dimensions  by  30  and  the  volume  of  each  unit  portion  will  be  increased  by  the  cube 
of  30.  The  attraction  of  each  unit  of  mass  on  the  given  point  will  vary  inversely 
with  the  square  of  the  distance  and  will  therefore  be  diminished  by  the  square  of  30. 
The  anomaly  will  consequently  increase  directly  with  the  dimensions,  provided  that 
the  density  remains  constant.  This  gives  the  basis  for  the  calculations  in  column  2, 
Table  XXI. 


3°4 


JOSEPH  BARRELL 


tions.     In  Table  XXI  the  attracting  mass  is  supposed  to  have  the 
form  of  a  vertical  cylinder.     With  a  given  anomaly  the  deficiency 

TABLE  XXI 

VERTICAL  CYLINDERS  GIVING  A  NEGATIVE  GRAVITY  ANOMALY  OF  o.ioo  DYNE  AT 
CENTER  OF  TOP  SURFACE  OF  CYLINDER 


I 

2 

3 

4 

5 

Diameter  

76  .  8    km. 

51  .2  km. 

102.4  km. 

51  .2  km. 

Depth.  . 

9.15  km. 

30.5  km. 

61  .0  km. 

61  .0  km. 

Density 

—  o  3i 

—  o  i? 

—  o  07 

—  o  12 

2.80—  Density  

2.49 

2.65 

2.73 

2.68 

Thickness  of  cylinder  of  same 
area  and  mass,  but  density 
2  67     

!i,o8o  m. 
3,550  ft. 

1,700  m. 
5,600  ft. 

1,700  m. 
5,600  ft. 

2,770  m. 
9,080  ft. 

Anomaly  per  100  feet  of  mass 
of  density  2.67  expanded 
to   depth   of   cylinder   as 
given  in  second  line  

0.0028  dyne 

0.0018  dyne 

0.0018  dyne 

o.oon  dyne 

TABLE  XXII 

SPHERES  GIVING  A  NEGATIVE  GRAVITY  ANOMALY  OF  o.  100  DYNE  AT  POINT  VERTI- 
CALLY ABOVE  ON  THE   SURFACE  OF  THE  EARTH 


I 

2 

3 

4 

5 

Diameter 

50.  km. 

100.  km. 

<co    km. 

100.  km. 

Depth  to  center 

25  .  km. 

50.  km. 

5      , 
32  .  km. 

64.  km. 

Density 

—  o  144 

—  0.072 

o 
—   0.236 

—  0.118 

2.80—  Density  

w  .   A*t*t 

2.66 

*  w  / 
2-73 

v    *O 

2.56 

2.68 

Length  of  polar  axis  of  oblate 

spheroid   of    same    equa- 

torial   section    and    same 

2,700  m. 

2,700  m. 

4,420  m. 

4,420  m. 

mass,  but  density  2  .  67  ... 
Anomaly    per    100    feet    of 

8,850  ft. 

8,850  ft. 

14,500  ft. 

14,500  ft. 

polar  axis  of  mass  at  den- 

sity 2.67  if  expanded  to 

diameter  of  sphere  

o.oon  dyne 

o.oon  dyne 

0.0007  dyne 

0.0007  dyne 

of  mass  will  be  least  if  the  cylinder  extends  from  the  station  down- 
ward instead  of  being  at  a  greater  depth.  Furthermore,  for  a 
given  volume  and  density  of  cylinder  the  gravitative  force  will 
vary  according  to  the  ratio  of  the  depth  to  the  diameter. 

Let#  =  depth 

Let  2R—  diameter 

Let  F  =  gravitative  force 


THE  STRENGTH  OF  THE  EARTH'S  CRUST       305 

Then  TrR2H=ihe  volume,  a  constant.     To  find  the  ratios  of  H 
to  R  which  give  maximum  attraction  for  unit  mass 

Let  R2H  =  i  and  solve  for  various  values  of  R  the  equation 


R2 

Several  solutions  are  as  follows: 

For  #  =  0.75,  R=I.I$;  F=(27rpy)  0.523 

H—i.oo,R  =  i.oo;  F  =  (277/07)  0.586 

H=i. 50,  R  =  o . 81 ;  F=  (277/07)  o . 609 

H  =  2.oo,  R  =  o.f/i;  F  =  (2Trpy)  0.586 

#  =  4.00,  jR  =  0-50;  F=(2irpy)  0.200 


This  shows  that  the  gravitative  force  is  a  maximum  for  a  cylinder 
of  constant  volume  and  mass  in  which  the  depth  varies  from  one- 
half  the  diameter  to  four- thirds  the  diameter.  The  force  varies 
but  slightly  between  those  limits.  The  cylinders  of  columns  3,  5, 
and  6,  of  Table  XXI,  lie  within  these  limits.  Thus  all  the  assump- 
tions thus  far  made  favor  the  minimization  of  the  negative  load 
which  produces  the  Seattle  anomaly. 

Taking  the  mean  density  of  the  outer  part  of  the  lithosphere 
as  2.80  it  is  seen  that  the  cylinder  of  column  2,  Table  XXI,  has  a 
density  below  that  of  the  lightest  rock-making  minerals  and  would 
require  the  existence  of  a  molten  magma  or  of  abnormal  pore  space 
to  great  depth.  It  may  therefore  be  eliminated  as  not  probable. 
Cylinders  3,  4,  and  5  show  densities  within  the  limits  of  granite, 
the  lightest  of  the  abyssal  igneous  rocks.  It  may  be  concluded, 
therefore,  that  the  deficiency  of  mass,  if  of  approximately  cylindrical 
form,  is  equivalent  to  a  negative  load  of  between  5,000  and  10,000 
feet  of  rock,  extending  over  a  distance  of  from  50  to  100  km.,  or 
a  somewhat  less  local  load  superposed  upon  a  broad  but  small 
regional  load  of  the  same  sign.  The  nature  of  the  assumptions 
has  been  such  that  we  may  conclude  with  confidence  that  the  Seattle 
anomaly  corresponds  to  at  least  5,000  feet  of  rock  and  may  reach 
a  considerably  higher  figure,  perhaps  10,000  feet.  Furthermore, 
Hayford's  unit  mass,  extending  to  the  areal  limits,  100  feet  thick 
and  density  2.67,  would  here  produce  an  anomaly  as  low  as  between 
o.ooio  and  0.0020  dyne. 


306  JOSEPH  BARRELL 

Instead  of  a  cylinder  suppose  the  mass  which  produces  the 
deficiency  of  gravity  to  approximate  more  to  the  form  of  a  sphere. 
The  results  are  shown  in  Table  XXII.  In  columns  2  and  3  the 
sphere  is  tangent  to  the  surface,  a  position  diminishing  the  mass  for 
a  given  anomaly.  In  columns  3  and  4  the  top  of  the  sphere  is 
7  and  14  km.  deep  respectively.  The  low  density  of  column  4 
shows  it  to  be  beyond  the  limiting  conditions.  The  load,  though 
negative  in  sign,  is  seen  to  be  equivalent  in  order  of  magnitude  to 
the  greater  volcanic  piles;  30  to  60  miles  in  diameter,  9,000  to 
14,000  feet  in  height  for  rock  of  density  2.67.  The  anomaly  pro- 
duced by  the  unit  mass  of  100  feet  thickness  and  density  2 . 67,  con- 
sidered here  as  100  feet  of  polar  diameter  for  a  spheroid  of  the  given 
horizontal  dimensions,  ranges  between  the  low  values  of  0.0007 
and  o.oon  dyne. 

From  a  consideration  of  these  two  tables  it  is  seen  that  the 
large  anomalies  require  either  a  variation  of  mass  equivalent  to 
as  much  as  5,000  feet  of  rock  extending  over  some  thousands  of 
square  miles  or  to  10,000  feet  of  rock,  more  or  less,  extending  over 
1,000  square  miles,  more  or  less.  These  tables  determine  the  order 
of  magnitude,  but  the  data  are  not  sufficient  to  permit  a  more 
accurate  solution  of  the  problem. 

Thus  this  detailed  examination  of  the  anomalies  in  the  region 
of  Seattle  shows  that  the  divisor  of  0.0030,  as  taken  by  Hayford 
and  Bowie,  or  0.0024,  as  considered  here  the  best  for  general  use, 
is  too  high  for  the  more  limited  areas  of  high  anomaly.  The  latter 
may  be  regarded  as  made  up  in  part  of  a  regional  portion  for  which 
the  divisor  of  0.0024  would  be  applicable  and  a  local  portion  for 
which  the  divisor  is  probably  not  over  0.0015.  As  a  mean  value, 
for  the  more  limited  areas  of  large  anomaly  the  amount  due  to  the 
unit  thickness  of  100  feet  of  rock  of  density  2 . 67  should  apparently 
not  be  taken  as  over  0.0020  dyne. 

In  forming  conceptions  as  to  the  uncompensated  vertical  stresses 
existing  widely  in  the  earth's  crust  it  is  important  to  know  the 
maximum  range  of  departures  from  the  mean  stress  as  well  as  the 
latter.  These  can  be  studied  well  in  Fig.  5.1  The  mean  of  four- 
teen maxima  of  defect  of  gravity  is  —0.033,  the  mean  for  eleven 

1  Fig.  5,  p.  153,  Part  II;   also  see  Hayford  and  Bowie,  pp.  107-8. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  307 

areas  of  excess  of  gravity  is  +0.034.  With  the  exception  of  the 
Seattle  stations  with  an  anomaly  of  —0.093,  none  reach  a  value 
of  0.060.  It  is  thus  seen  that  the  average  notable  maximum  is 
not  far  from  twice  the  mean  anomaly.  Even  by  using  a  uniform 
divisor  of  0.0024  or  0.0030  to  convert  anomalies,  regional  depar- 
tures of  load  amounting  to  1,300  or  1,500  ft.  over  areas  of  several 
square  degrees  are  found  to  be  not  uncommon.  Over  smaller 
areas  the  loads  rise  to  about  three  times  the  mean,  and  at  Seattle 
to  five  times  the  mean.  These  figures  of  course  do  not  measure 
simply  the  elevations  or  depressions  of  uncompensated  erosion 
features.  On  the  contrary,  if  the  hills  and  valleys  be  imagined 
as  smoothed  out,  then  the  resulting  mean  surface  would  be  out  of 
isostatic  equilibrium  in  the  same  direction  over  distances  amount- 
ing frequently  to  hundreds  of  kilometers  and  attaining  maximum 
departures  too  low  or  too  high  over  smaller  areas  by  these  figures. 

But  an  inspection  of  the  contour  map  of  gravity  anomalies 
(Fig.  5)  shows  that  the  large  anomalies,  those  of  0.040  dyne  or 
above,  are  all  located  by  Hay  ford  and  Bowie  as  centers  of  maximum 
anomaly,  though  the  nearest  adjacent  stations  average  as  much 
as  100  miles  distant.  Between  the  widely  spaced  stations,  the 
anomaly  gradients  are  gentle.  But  where  the  stations  form  a 
series  closer  together,  as  that  from  the  city  of  Washington  to  New 
York  City,  the  gradients  are  seen  to  be  steeper  and  more  irregular. 
It  is  to  be  presumed  that  a  further  multiplication  of  stations  would 
show  increased  complexity  over  the  whole  country  and  reveal 
maxima  higher  than  those  now  recorded.  The  value  of  the  mean 
anomaly  without  regard  to  sign  should  furthermore  increase  some- 
what through  the  discovery  of  additional  areas  of  maximum 
value.  Areas  of  regional  positive  or  negative  anomaly  would 
persist  in  something  of  their  present  size,  but  within  broad  areas 
of  anomaly  of  one  sign  should  be  discovered  smaller  areas  of  oppo- 
site sign  which  are  now  unknown.  Upon  the  completion  of  such 
a  detailed  survey  the  high  anomaly  of  Seattle  would  not  appear 
so  exceptional  as  it  does  at  present. 

The  chart  of  the  residuals  of  Solution  H1  shows  within  the 
larger  areas  of  like  deflection  of  the  vertical  many  large  and  sharp 

1  Illustration  No.  3,  Hayford,  Supplementary  Paper,  Bowie,  Illustration  No.  5. 


308  JOSEPH  BARRELL 

variations  in  value  and  in  direction.  The  resultants  of  the  plotted 
arrows  point  toward  the  centers  of  exceptional  mass  and  their 
rapid  changes  in  value  and  direction  point  toward  the  existence  of 
many  comparatively  shallow  masses.  The  epicentral  points  above 
such  masses  are  those  where  the  gravity  anomaly,  Fv,  is  at  a  maxi- 
mum. If  a  hidden  mass  may  be  regarded  as  approaching  a  spheri- 
cal form  and  has  its  center  at  depth  D,  the  following  relations  exist 
between  the  value  of  the  gravity  anomaly  and  the  distance  x 
from  the  epicenter: 

Fv  =  Maximum  f or  x  =  o .  oo  D 

7fy=  .75  max.      "   x  =  o.46D 

^=.50  max.      "   x  =  o.^D 

7fy=  .25  max.      "  x  =  i.2$D 

If,  for  example,  an  approximately  spherical  mass  has  its  center  at 
a  depth  of  32km.,  .005  of  the  earth's  radius,  the  anomaly  Fv 
will  fall  to  half -value  at  a  distance  of  25  km.  from  the  epicenter. 
If  the  center  is  64  km.  deep,  the  anomaly  will  fall  to  half-value 
at  50  km.  from  the  epicenter.  Between  stations  located  100  km. 
apart  by  far  the  greater  number  of  real  maxima  would  be  missed, 
and  in  so  far  as  they  depended  upon  masses  in  the  upper  half  of  the 
zone  of  compensation  the  indicated  maxima  would  at  most  places 
be  less  than  one-half  the  real  maxima. 

The  stresses  acting  within  the  crust  owing  to  excesses  or  defi- 
ciencies of  mass  are  not  so  concentrated  and  therefore  not  quite  so 
great  as  if  those  abnormalities  of  mass  existed  as  surface  loads  of 
rock  of  density  2.67  in  the  manner  imagined  for  the  interpretation 
of  anomalies.1  Nevertheless  to  gain  a  conception  of  the  meaning 
of  the  gravity  anomalies,  imagine  the  present  compensated  topog- 
raphy to  be  smoothed  out  to  sea-level  and  the  variations  of  mass 
away  from  isostatic  equilibrium  to  become  variations  of  volume 
upon  its  surface.  The  anomaly  contours  will  then  become  topog- 
raphy contours,  the  line  of  zero  anomaly  will  become  the  datum 
plane.  The  values  in  mass  to  be  assigned  to  the  successive  anomaly 
contours  can  only  be  given  in  mean  figures.  It  has  been  shown 
however  in  Part  III  that  balanced  vertical  irregularities  of  density 

1  The  relations  of  mass  and  its  distribution  to  the  resulting  stresses  will  be  con- 
sidered in  a  later  part. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST 


309 


do  not  play  a  large  part  in  causing  gravity  anomalies.  It  will 
be  shown  later  that  neither  can  nucleal  heterogeneity  below  the 
200  to  300  km.  level  of  isostatic  compensation  account  for  a  large 
part.  The  anomalies  represent  in  greater  part  real  departures  from 
isostasy  and,  as  shown  in  this  section,  the  limited  areas  of  high 
anomaly  are  to  be  interpreted  as  implying  on  the  whole  a  local 
load  in  higher  ratio  to  anomaly  than  do  the  broad  areas  of  anomaly. 
The  average  relation  thought  to  exist  is  shown  then  in  the  following 
table: 

TABLE  XXIII 

AN  ESTIMATED  AVERAGE  RELATION  OF  ANOMALY  CONTOURS 

TO  CONTOURS  OF  EQUIVALENT  ROCK  MASSES 

OF  DENSITY  2.67 


Anomaly  Contour,  Posi- 
tive or  Negative 

Assumed  Divisor  for  100 
Feet  of  Rock  upon  a 
Level  Surface 

Equivalent  Contour  in 
Feet,  Positive  or 
Negative 

.020 

.0025 

800 

.040 

.0023 

I7OO 

.060 

.0020 

3000 

.080 

.0018 

4500 

.100 

.0016 

6300 

Upon  conversion  of  a  detailed  anomaly  map,  if  such  existed, 
into  the  equivalent  topographic  contour  map  by  means  of  such 
ratios  as  those  given  in  Table  XXIII,  the  whole  United  States  with 
its  compensated  topography  previously  smoothed  out  to  sea-level 
would  be  reconverted  into  a  roughly  mountainous  country  with  no 
notable  distinction  between  what  are  now  the  central  plains  and 
mountainous  border  regions  of  the  continent.  On  to  broad  plateaus 
or  basins  upward  of  1,000  feet  from  the  mean  elevation  would  be 
added  higher  elevations  and  depressions.  The  extreme  differ- 
ential relief  would  probably  be  in  the  neighborhood  of  two  miles 
though  the  average  departure  without  respect  to  sign  from  the 
mean  surface  of  the  geoid  would  probably  be  between  800  and 
1,000  feet.  Though  everywhere  as  irregular  as  a  mountainous 
country,  there  would  be  little  or  no  relief  of  the  mean  level  of  this 
hypothetical  surface  above  the  ocean  bottoms  and  no  such  broad 
and  high  masses  as  the  Cordilleran  plateaus  would  remain  within 


3 10  JOSEPH  BARRELL 

its  limits.  The  major  relief  of  the  continent  above  the  ocean  bot- 
toms would  be  about  nine-tenths  eliminated  and  the  mean  eleva- 
tion of  all  areas  as  great  as  several  hundred  miles  in  width  would 
be  reduced  to  a  small  figure.  This  is  the  effect  of  isostasy.  But 
within  these  unit  areas  which  measure  the  limits  of  regional  com- 
pensation would  everywhere  rise  a  rolling  mountainous  surface. 

Imagine  the  hypothetical  surface  as  broad  as  the  United  States, 
concealed  from  view  by  an  impenetrable  envelope  of  cloud,  and 
aerial  explorers  to  sink  a  sounding  line  to  this  invisible  land  at  124 
places  chosen  at  random.  The  resulting  contour  map  compiled 
from  these  soundings  would  yield  a  much  smoothed  and  flattened 
surface  such  as  is  shown  in  the  contour  map  of  gravity  anomalies. 
Many  of  the  soundings  taken  really  on  mountain  slopes,  because 
they  were  the  highest  of  those  made,  might  be  casually  interpreted 
as  located  on  mountain  peaks.  The  latter,  standing  sharp  and  high, 
would  be  missed  save  for  an  occassional  lucky  chance  of  the  sound- 
ing line. 

Interpreted  in  terms  of  weights  and  stresses,  it  is  seen  that  even 
the  parts  of  the  continent  appearing  to  the  eye  as  plains  long  in 
geologic  quietude  really  conceal  within  them  strains  as  .great  as 
those  imposed  by  the  weight  of  mountains.  That  these  great 
strains  have  been  born  for  geologic  ages,  in  many  localities  probably 
from  the  Archeozoic,  gives  a  surprising  conception  of  an  enduring 
rigidity  and  elasticity  of  the  crust  wholly  at  variance  with  certain 
current  doctrines  regarding  the  weakness  of  this  zone.  It  is  not 
here  found  to  be  a  failing  structure. 

On  p.  8 1  Hayford  and  Bowie  give  the  new-method  anomalies 
for  sixteen  stations  not  in  the  United  States.  An  abstract  is  given 
below  of  the  greater  anomalies  from  that  table  with  the  addition 
of  Seattle.  The  thickness  of  stratum  taken  as  corresponding  to 
the  anomaly  is  also  added.  This  thickness,  if  the  compensation 
is  uniform  with  depth,  measures  the  distance  by  which  the  earth's 
surface  is  out  of  isostatic  equilibrium  at  those  points.  A  plus  sign 
indicates  an  excess  of  mass  and  a  consequent  tendency  to  sink, 
resisted  by  rigidity;  a  minus  sign  a  defect  of  mass  and  therefore 
the  existence  of  an  upward  strain. 

The  divisor  0.003  dyne  °f  anomaly,  taken  as  the  equivalent 


THE  STRENGTH  OF  THE  EARTH'S  CRUST 


of  100  feet  of  abnormal  mass  of  density  2.67,  is  Hayford's  figure. 
As  previously  discussed  it  is  thought  to  minimize  too  much  the 
thickness  of  equivalent  rock.  It  is  given,  however,  for  compari- 
son with  the  column  derived  from  the  use  of  0.0024  as  a  divisor. 
This  is  regarded  as  a  better  average  figure,  but  for  some  cases  at 
least,  as  shown  for  Seattle,  this  also  may  give  too  low  a  result. 

TABLE  xxiv 


NUMBER  AND  NAME  OF  STATION 

ELEVATION  IN 
METERS 

NEW-METHOD 
ANOMALY 

THICKNESS  OF  EQUIVALENT 
STRATUM  ON  LAND  IN  FEET 

0.003  Anomaly 
=  100  feet 

0.0024  Anomaly 
=  100  feet 

2.  Tonga  plateau,  Hecker, 
at  sea  

—  2,700 
-6,500 

+3,98l 
+       21 
+       19 

+0-255 
-    .184 

+    .183 
+   .no 
+   .146 
-0.095 

+8,500 
-6,130 

+6,130 
+3,670 
+4,870 
—  3^70 

+  10,625 
-    7,660 

+   7,660 
+  4,590 
+  6,090 
-   3,960 

4.  Tonga  deep,  Hecker,  at 
sea   ... 

9.  Mauna  Kea,  Hawaiian 
Islands 

10.  Hachinohe,  Japan  
13.  Sorvaagen,  Norway.  .  .  . 
Seattle,  United  States  .  . 

It  is  seen  that  the  excesses  of  mass  indicated  for  Mauna  Kea 
and  at  Sorvaagen  are  each  comparable  in  equivalent  thickness  and 
extent  to  the  maximum  thickness  of  the  Niger  Delta  if  measured 
by  rock  upon  land,  5,450  feet.  The  departures  from  equilibrium 
at  Hachinohe,  Japan,  and  Seattle,  United  States,  are  comparable 
in  thickness  and  area  to  the  burden  of  the  Nile  Delta,  the  weight 
in  air  of  3,600  to  4,200  ft.  of  rock.  In  weight  as  in  area,  therefore, 
these  deltas  are  seen  to  impose  burdens  on  the  crust  no  greater  than 
are  found,  by  means  of  geodetic  observations,  to  exist  in  certain 
other  regions  where  geologic  evidence  had  not  revealed  them. 
The  accuracy  of  Hecker 's  method  for  determining  the  intensity  of 
gravity  at  sea  has  been  called  into  question  by  Bauer1  so  that,  until 

1  "On  Gravity  Determinations  at  Sea,"  Amer.  Jour.  Science  (4),  XXXI  (1911), 
1-18.  "Hecker's  Remarks  on  Ocean  Gravity,"  Amer.  Jour.  Science  (4),  XXXIII 
(1912),  245,  248. 


312  JOSEPH  BARRELL 

this  question  is  settled  by  geodesists,  equal  weight  should  perhaps 
not  be  attached  to  the  figures  given  for  the  departures  from  isostasy 
shown  over  the  Tonga  plateau  and  Tonga  deep.  Neither  is  the 
area  of  these  departures  known,  though  the  areas  of  the  plateau 
and  deep  are  large.  These  regions  are  seen,  however,  to  indicate 
considerably  higher  departures  from  isostasy  than  the  measure- 
ments determined  from  the  deltas  of  the  Nile  and  Niger.  The 
latter,  therefore,  perhaps  do  not  measure  the  full  strength  of  the 
crust. 

Major  H.  L.  Crosthwait  has  applied  Hay  ford's  methods  to  the 
investigation  of  isostasy  in  India.1  The  residuals  of  the  deflections 
of  the  vertical  serve  as  a  measure  of  the  degree  of  compensation 
existing  in  the  United  States  as  compared  in  India  and  are  as  fol- 
lows: 

UNITED  STATES  OF  AMERICA 

Group  S.E.,  mean  residual ; —  0^74 

Group  N.E.,  mean  residual —  i .  04 

Group  Central,  mean  residual —  i .  66 

Group  W.,  mean  residual —  4. 02 

INDIA 

Region  No.  i,  Himalaya  Mountains,  mean  residual —  i6T 

Region  No.  2,  Plains  at  foot  of  Himalaya  Mts.,  mean  residual —  2. 

Region  No.  3,  N.E.,  mean  residual +  8. 

Region  No.  4,  Central,  mean  residual -+-  5 . 

Region  NO.  5,  N.W.,  mean  residual -f  4. 

Region  No.  7,  W.,  mean  residual —  3 . 

Region  No.  8,  E.,  mean  residual —  z. 

Region  No.  9,  S.,  mean  residual -f  i . 

It  is  seen  that  the  residuals  average  several  times  as  great  in 
India  as  in  the  United  States,  which  leads  him  to  conclude  that 
"  Speaking  generally  it  would  appear  that  isostatic  conditions  are 
much  more  nearly  realized  in  America  than  in  India,  i.e.,  if  we 
are  to  take  the  smallness  of  the  residuals  as  an  indication  of  the 
completeness  of  isostatic  compensation."2  Colonel  Burrard,  utiliz- 
ing the  Hayfordian  computations,  points  out  the  existence  of  zones 

1  Professional  Paper  No.  13,  Survey  of  India,  1912. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  313 

in  India  where  the  deflections  of  the  plumb-line  are  actually  in 
opposition  to  the  directions  called  for  by  isostasy.1  The  major 
elements  of  the  relief,  the  Himalayas,  the  plateau  of  India,  and  the 
surrounding  ocean  basins  are  of  course  largely  compensated,  but 
these  figures  show  that  in  detail  the  hypothesis  of  complete  isostasy 
is  very  far  from  the  truth.  Crosthwait  suggests  that  the  explana- 
tion for  the  difference  between  the  United  States  and  India  probably 
lies  in  the  magnitude  of  the  recent  upheavals  of  the  crust  in  that 
part  of  the  globe.  Nevertheless  such  upheavals  cannot  exceed  the 
strength  of  the  crust,  and  in  India,  therefore,  perhaps  may  be  better 
observed  than  in  the  United  States  the  maximum  strains  which  the 
earth  is  competent  to  endure. 

It  may  be  concluded,  therefore,  that  the  convergence  of  geodetic 
evidence  shows  the  crust  to  be  competent  to  sustain  loads  measured 
by  the  weight  of  several  thousand  feet  of  rock  extending  over 
circular  areas  some  tens  of  thousands  of  square  miles  in  area. 
This  is  a  measure  of  crustal  strength  twenty,  fifty,  or  even  a  hundred 
fold  greater  than  that  advanced  in  recent  years  by  the  leading  cham- 
pions of  high  isostasy. 

FURTHER  GEODETIC  WORK,  NEEDED  FOR  GEOLOGIC  PROBLEMS 

It  has  been  the  intention  in  the  preceding  analysis  to  show  two 
things:  first,  that  the  data  set  forth  by  Hayford  and  Bowie  are  of 
great  value  to  geology  and  establish  new  methods  of  research, 
but,  second,  that  the  difficulties  inherent  in  the  observations  and 
their  mathematical  treatment,  and  the  fewness  of  the  stations  in 
comparison  with  the  heterogeneity  of  the  earth,  are  such  that  the 
conclusions  from  the  geologic  study  of  deltas  in  the  first  part  of 
this  paper  are  as  convincing  and  perhaps  as  accurate  as  the  present 
results  of  the  geodetic  studies.  The  latter,  however,  opens  for 
the  whole  earth  a  field  of  investigation  which  the  geologic  evidence 
covers  very  locally  and  imperfectly,  a  world-wide  field  which  should 
be  pursued  for  the  geologic  as  much  as  for  the  geodetic  bearings. 

By  means  of  the  divining  rods  of  pendulum  and  plumb-line  the 
heterogeneities  of  mass  and  the  loci  of  strain  in  the  outer  crust  of 

1  "On  the  Origin  of  the  Himalaya  Mountains,"  Professional  Paper  No.  12,  Survey 
of  India,  1912. 


314  JOSEPH  BARRELL 

the  earth  should  be  sought  out  and  measured  in  detail.  For  this 
work  it  would  seem  that  many  new  stations  would  have  to  be 
established;  in  groups  so  as  to  reduce  the  errors  of  each  locality; 
in  sets  so  as  to  attack  particular  phases  of  the  problems.  For 
example,  it  would  appear  that  gravity  stations  should  be  located 
in  pairs  close  together  and  of  as  great  a  difference  in  elevation  as 
possible.  Certain  stations  should  be  located  within  areas  of 
plateaus  spared  by  circumdenudation,  such  as  the  Cumberland  and 
Allegheny  plateaus;  others  should  be  located  in  the  broad  erosion 
basins.  Deflection  stations  should  be  located  on  the  lines  separat- 
ing regions  of  erosion  from  those  of  circumdenudation,  and  also  on 
the  lines  separating  areas  of  upwarp  from  those  of  downwarp.  A 
network  should  inclose,  finally,  all  centers  of  marked  gravity 
anomaly  or  topographic  deflection.  Such  an  increase  in  the  number 
of  stations  would  permit  the  introduction  of  simple  hypotheses  of 
variable  depth  and  rate  and  regional  limits  of  compensation.  But 
such  an  extensive  program  is  within  the  reach  only  of  some  research 
institution.  It  needs  the  co-operation  of  geologists  and  geodesists. 
The  location  of  stations  with  respect  to  surface  features  and  their 
geologic  history  should  be  controlled  by  the  geologist.  The  density 
of  the  rocks  to  the  limits  exposed  by  the  structure  should  also  be 
determined  by  him.  The  geodesist,  on  the  other  hand,  should 
seek  out  the  hidden  heterogeneities  in  the  crust  and  guide  the 
details  of  the  work. 

[To  be  continued] 


VOLUME  XXII  NUMBER  5 


THE 


JOURNAL    OF   GEOLOGY 

JULY-AUGUST  1914 


THE  STRENGTH  OF  THE  EARTH'S   CRUST 


JOSEPH  BARRELL 
New  Haven,  Connecticut 


PART   V.    THE   DEPTH   OF   MASSES    PRODUCING   GRAVITY 
ANOMALIES  AND  DEFLECTION  RESIDUALS 

INTRODUCTION  AND  SUMMARY     ...          .          .  ...     441 

SECTION  A1 

DEVELOPMENT  OF  CRITERIA  FOR  SPHEROIDAL  MASSES    ....  446 

Separation  of  Lithospheric  from  Centrospheric  Outstanding  Masses  446 

Influence  of  Centrospheric  Heterogeneity 448 

Distribution  of  Surface  Forces  for  Centrospheric  Spheres  .  .  449 
Influence  of  Spheres  within  the  Zone  of  Compensation  .  .  .452 
Influence  of  Sum  of  Intersecting  Spheres  Approximately  Equivalent 

to  Spheroids    .       .       .        .               453 

Distinctive  Effects  of  Individual  Spheres  and  Spheroids  .  .  .  458 
Depths  of  Spheres  Whose  Epicenters  Are  Not  on  the  Line  of 

Traverse       .       ... .     .       ._.....       .        .        .  463 

INTRODUCTION  AND    SUMMARY 

The  fact  that  the  observed  deflections  of  the  vertical  are  on 
the  average  only  one-tenth  as  large  as  the  computed  effects  of  the 
topographic  relief,  computed  on  the  assumption  of  uniform  density 
throughout  each  earth  shell,  shows  that  the  densities  of  the  crust 

1  Section  B  of  Part  V,  on  the  applications  of  the  criteria  to  determine  the  limiting 
depths,  forms,  and  masses  of  the  excesses  and  defects  of  density,  will  be  published  in 
the  following  number  of  this  Journal. 

Vol.  XXII,  No.  5  441 


442  JOSEPH  BARRELL 

are  not  uniform,  but  in  a  broad  way  are  balanced  against  the  relief. 
This  is  the  proof  that  a  condition  prevails  of  approximate  regional 
isostasy.  As  the  relief  of  the  globe  is  highly  variable,  the  densities 
in  the  lithosphere  are  therefore  within  certain  limits  also  highly 
variable.  But  on  the  other  hand,  the  existence  of  gravity  anomalies 
and  deflection  residuals  indicates  that  the  variations  in  density  are 
not  completely  in  accord  with  the  demands  of  the  hypothesis  which 
postulates  local  compensation  of  the  topography  uniformly  dis- 
tributed to  a  uniform  depth,  nor  apparently  with  any  other  simple 
hypothesis.  These  quantities  measure  the  differences  between 
the  hypothesis  and  the  facts  of  nature.  Let  the  density  variations 
beyond  those  required  to  balance  the  topography  vertically  above 
them  be  called  the  outstanding  excesses  or  defects  of  density  and 
the  masses  which  they  represent  be  called  the  outstanding  masses. 
It  is  fundamental  to  the  problems  of  the  strength  of  the  crust, 
and  a  system  of  geologic  dynamics  in  accord  with  that  strength,  to 
determine  the  depth,  form,  and  weight  of  these  outstanding  masses. 
Do  they  belong  to  the  centrosphere  or  to  the  lithosphere  ?  As 
Gilbert  has  noted,  if  they  are  to  be  referred  to  the  centrosphere 
they  do  not  imply  any  imperfection  of  isostasy  nor  any  competence 
for  stress  within  the  crust.  Or,  if  they  exist  in  the  lithosphere,  the 
zone  of  compensation,  but  are  balanced  vertically  in  the  same 
column  by  other  masses  of  opposite  sign,  this  arrangement  will 
produce  local  strains  within  the  crust  but  not  tend  to  flex  the  crust 
as  a  whole.  Neither  in  this  case,  therefore,  would  they  measure 
departures  from  perfect  isostasy.  As  following  questions,  are  the 
imperfections  of  isostasy  small  and  local,  and  the  residuals  and 
anomalies  the  summation  of  many  scattered  effects  ?  Or,  on  the 
contrary,  are  there  notable  regional  departures  from  the  conditions 
of  solid  flotation  which  measure  a  very  appreciable  rigidity  of  the 
crust  ?  If  so,  to  what  extent  are  these  regional  outstanding  masses 
related  to  the  mountains,  valleys,  and  deltas  in  process  of  evolu- 
tion under  the  present  cycle  of  surface  activities;  producing  a 
progressive  unbalancing  possibly  being  slowly  restored  toward 
balance  by  a  viscous  undertow  ?  To  what  extent  are  the  departures 
from  isostasy  due  to  variable  composition  and  density  of  igneous 
intrusions  dating  back  to  earlier  geologic  ages,  perhaps  never  in 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  443 

isostatic  balance,  and  supported  permanently  and  rigidly  by  the 
strength  of  the  crust  ? 

These  problems  have  been  studied  in  the  last  three  parts  of  this 
article  by  means  of  the  evidence  presented  by  Hay  ford  and  Bowie, 
but  that  evidence  has  not  been  used  to  solve  the  depth  of  the  out- 
standing masses.  Yet  it  is  seen  that  all  of  the  aspects  just  enu- 
merated are  bound  up  in  that  factor.  It  is  especially  this  problem 
of  the  depth  and  the  consequent  areal  extent  and  mass  of  the  units 
which  produce  the  residuals  of  the  deflection  and  gravity  observa- 
tions which  is  attacked  in  this  part.  It  is  necessary  for  this  investi- 
gation to  enter  into  a  study  of  the  complex  relations  between  the 
anomalies  and  residuals  which  depend  upon  the  depth  and  form 
of  masses.  It  is  a  subject  upon  which,  so  far  as  the  writer  is 
aware,  but  little  has  been  done,  so  that  about  half  of  this  chapter, 
published  as  Section  A,  consists  of  a  study  of  these  relations  pre- 
liminary to  their  application. 

For  facility  of  mathematical  treatment  the  individual  outstand- 
ing masses  must  be  regarded  as  equivalent  to  spheres,  spheroids,  or 
cylindrical  disks,  either  as  units  or  as  aggregates.  If  only  the 
epicenter  (the  point  on  the  surface  vertically  above  the  center)  of 
the  disturbing  mass  is  determinable  and  the  deflections  at  two  or 
three  points  on  one  side  of  it,  then  the  mass  may  be  most  simply 
interpreted  as  a  sphere;  since  the  mass  of  a  sphere  acts  as  if  con- 
centrated at  its  center.  With  fuller  observations  a  close  approxi- 
mation to  the  depth  of  the  mass  and  a  less  close  approximation 
to  its  form  and  density  may  be  made.  The  first  problem  then  is 
to  determine  the  epicenter  of  the  outstanding  mass  and  its  depth, 
using  for  this  purpose  the  nature  of  the  anomalies  and  residuals. 
For  a  sphere  beneath  a  plane  surface  it  is  shown  that  the  value  of 
the  maximum  gravity  anomaly  at  the  surface  is  2.6  times  the 
value  of  the  maximum  deflection  residual,  both  being  measured  in 
the  same  units  of  force.  The  former  occurs  vertically  over  the 
abnormal  spherical  mass,  that  is,  at  the  epicenter.  The  latter 
occurs  at  a  horizontal  distance  from  the  epicenter  equal  to  70  per 
cent  of  the  vertical  depth  to  the  center  of  mass.  Oblate  spheroids 
and  broad  cylindrical  disks  with  vertical  axes  and  the  same  depth 
of  center  as  the  sphere  give  maximum  deflections  at  greater 


444  JOSEPH  BARRELL 

distances  from  the  epicenter.  The  curves  of  the  deflection  force 
for  these  forms,  especially  the  spheroidal  forms,  resemble  somewhat 
closely  those  given  by  deeper  spheres  of  greater  mass.  If  the  out- 
standing masses  are  in  reality  horizontally  extended  the  interpre- 
tation of  the  deflection  residuals  as  due  to  spherical  masses  assigns 
to  their  centers  in  consequence  too  great  a  depth.  If,  however, 
the  masses  have  the  forms  of  vertical  prolate  spheroids  or  vertical 
elongate  cylinders,  the  interpretation  as  spheres  will  give  too  shallow 
a  depth.  The  ratios  of  the  maximum  anomalies  to  the  maximum 
deflections  constitute  a  criterion  to  show  whether  the  masses  depart 
from  spheres  by  the  spreading-out  of  their  substance  in  a  horizontal 
plane  or  along  a  vertical  ^,xis. 

In  Section  B  the  outstanding  masses  are  shown  in  some  cases 
to  be  horizontally  extended  in  form  and  this  is  thought  for  the 
larger  masses  to  be  a  rather  general  relationship;  that  is,  the  verti- 
cal thickness  is  much  less  than  the  length  or  breadth.  Conse- 
quently the  interpretation  as  spheres  gives  maximum  limits  to  the 
depth. 

A  general  inspection  of  the  geodetic  data  as  well  as  a  detailed 
study  of  a  certain  test  region  shows  that  the  smaller  disturbing 
masses  have  their  centers  in  the  outer  third  of  the  zone  of  com- 
pensation; that  is,  within  40  km.  of  the  surface.  This  result  is 
to  be  expected,  since  similar  small  masses  at  greater  depth  would 
not  exert  a  notable  effect  because  their  gravitative  force  varies 
inversely  with  the  square  of  the  distance.  But  evidence  of  more 
significance  is  found  in  regard  to  the  larger  centers  of  outstanding 
mass  not  related  to  topography.  These  also  are  found,  in  so  far 
as  they  have  been  investigated,  to  be  situated  in  the  outer  third 
of  the  zone  of  compensation.  Yet  these  masses  are  capable  of 
showing  notable  effects  to  distances  of  from  100  to  150  km.  If 
they  were  situated  at  any  depth  within  the  zone  of  compensation 
they  would,  therefore,  betray  both  their  existence  and  their  depth. 
The  greater  departures  from  isostasy  appear,  therefore,  to  be  really 
absent  from  the  deeper  parts  of  the  lithosphere. 

Centrospheric  heterogeneity,  if  present,  would  require  greater 
masses  in  order  to  show  surface  effects.  But  no  such  effects  are 
noted.  In  so  far  as  they  may  be  existent,  they  are  largely  masked 


THE  STRENGTH  OF  THE  EARTH'S  CRUST       445 

by  the  more  important  attractions  of  superficial  masses  and  hidden 
by  the  indeterminate  nature  of  much  of  the  present  data.  There- 
fore centrospheric  heterogeneity  is  not  a  hypothesis  which  can  be 
used  to  account  for  the  apparent  departures  from  isostasy.  It 
can  be  at  most  only  a  very  secondary  factor. 

In  the  last  topic  of  Section  B  is  discussed  the  relation  of  the 
depth  of  outstanding  masses  to  the  various  hypotheses  regarding 
the  distribution  of  compensation.  The  hypothesis  of  local  com- 
pensation, as  is  perceived  to  a  certain  extent  by  those  who  have 
used  it,  is  in  error  in  supposing  that  variations  in  density  correspond 
to  every  topographic  feature  and  extend  uniformly  to  the  bottom 
of  the  zone.  But  these  errors,  whether  they  be  small  or  great,  are 
so  spread  out  in  depth  and  their  centers  of  attraction  are  conse- 
quently so  far  removed  from  the  surface  that  they  have  little  effect 
on  the  geodetic  observations.  Especially  is  this  true  in  compari- 
son with  those  large  and  concentrated  outstanding  masses  due  to 
batholithic  invasion  or  other  causes  which  are  found  to  exist  at 
moderate  depths  in  the  outer  crust.  The  reasons  then  why  the 
deflection  residuals  and  gravity  anomalies  appear  to  show  so  little 
relation  to  local  surface  relief  and  larger  physiographic  provinces 
are  threefold:  in  part  because  of  a  regional  compensation;  in  part 
because  the  hypothesis  of  local  compensation  as  here  shown  masks 
the  error  contained  in  the  assumption  of  perfect  and  local  isostasy; 
in  part  because  for  many  regions  the  ancient  heterogeneities  of 
mass  hidden  within  the  crust  seem  in  reality  to  be  greater  than  the 
heterogeneities  of  mass  visible  at  the  surface  in  topographic  form 
and  created  by  present  gradational  actions. 

The  results  of  this  chapter  converge  with  the  lines  of  evidence 
previously  considered  and  confirm  them  in  showing  considerable 
defects  from  isostasy  for  areas  which  are  100  km.  or  more  in 
radius.  This  confirmation  is  to  be  expected,  since  it  would  be 
indeed  remarkable  if  a  crust,  competent  to  carry  such  loads  as  the 
geologic  evidence  from  erosion  and  sedimentation  shows  to  be 
imposed,  should  give  geodetic  evidence  of  fairly  local  and  nearly 
perfect  adjustment  between  the  topographic  forms,  developed  by 
present  external  processes,  and  the  variations  in  density  imposed 
by  past  internal  forces. 


446  JOSEPH  BARRELL 

SECTION  A 
DEVELOPMENT  OF  CRITERIA  FOR  SPHEROIDAL  MASSES 

Separation  of  litho  spheric  from  centrospheric  outstanding  masses. — 
Let  the  zone  of  compensation  be  regarded  as  the  boundary  of  the 
lithosphere.  At  its  bottom  consider  to  exist  a  zone  in  which  that 
lateral  flowage  takes  place  which  is  necessary  for  movements 
of  isostatic  readjustment  and  the  maintenance  through  geologic 
time  of  a  condition  of  more  or  less  complete  isostasy.  Below  it 
is  the  inner  and  more  rigid  core  of  the  earth,  the  centrosphere. 
Let  those  excesses  or  defects  of  density  above  the  zone  of  isostatic 
flow  which  are  not  in  accord  with  the  isostatic  compensation  of  the 
topography  be  designated  for  convenience  as  lithospheric  outstand- 
ing masses.  Let  all  heterogeneities  of  density  within  any  earth 
shell  below  the  zone  of  isostatic  flow  be  called  centrospheric  out- 
standing masses. 

In  his  recent  paper  on  the  "Interpretation  of  Anomalies  of 
Gravity,"1  Gilbert  calls  attention  to  the  fact  that  if  abnormalities 
of  density  exist  below  the  zone  of  compensation  they  will  pro- 
duce anomalies  of  gravity  without  these  signifying  real  departures 
from  isostasy.  This  is  a  very  necessary  addition  to  the  theory 
of  the  cause  of  gravity  anomalies  and  deflection  residuals.  As  a 
test,  Gilbert  has  calculated  the  influence  of  a  right  cylinder  with 
vertical  axis,  of  density  ±0.025,  with  height  and  radius  each  equal 
to  122  km.,  whose  upper  surface  is  at  a  depth  of  122  km.,  thus 
reaching  up  to  the  bottom  of  the  zone  of  compensation  as  given  by 
Solution  H.  Such  a  cylinder  would  give  a  maximum  anomaly  of 
±0.023  dyne  at  the  epicenter,  a  quantity  of  the  same  order  of 
magnitude  as  the  mean  anomaly  for  the  United  States,  0.018  or 
0.020  dyne. 

In  the  application  of  this  test  to  the  earth  it  would  appear, 
however,  that  two  things  should  be  noted.  First,  to  account  for 
the  mean  anomaly  of  0.020  dyne  the  centrospheric  masses  would 
have  to  be  several  times  as  great  as  this  cylinder,  even  for  this  depth 
of  122  km.  to  the  top  surface,  since  the  maximum  value  of  the 
anomaly  occurs  at  the  epicenter  of  the  mass,  and  for  a  cylinder  of 

1  Professional  Paper  SjC,  U.S.  Geol.  Survey,  1913,  pp.  35,  36. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  447 

the  form  postulated  falls  off  rapidly  with  increasing  horizontal 
distance  from  the  epicenter.  Second,  the  test  mass  has  been  taken 
as  contiguous  to  the  zone  of  compensation  above  and  with  that 
limited  depth  given  by  the  hypothesis  of  uniform  compensation. 
This  gives  it  greater  effect  according  to  the  law  of  inverse  squares, 
but  postulates  either  an  indefinitely  thin  zone  of  isostatic  flow  at  the 
bottom  of  the  zone  of  compensation  or  a  capacity  in  a  thicker  zone 
of  weakness  to  maintain  within  itself  heterogeneities  of  density 
similar  to  those  of  the  lithosphere  above  and  the  centrosphere 
beneath.  It  is  thought  by  the  present  writer  that  a  more  probable 
presumption  is  that  the  centrospheric  heterogeneities  which  may 
exist  are  distinctly  deeper  than  122  km.  and  separated  from  the 
lithospheric  outstanding  masses  by  a  thick  zone  which  yields  to 
broad  inequalities  of  pressure  either  upon  it  or  within  it  and  there- 
fore is  incapable  of  maintaining  notable  inequalities  of  mass  in 
this  shell, 

The  reasons  for  this  preliminary  hypothesis  are  briefly  as  fol- 
lows: The  depth  of  compensation  seems  to  be  variable  and  to  extend 
in  some  regions  to  as  much  as  300  km.,  even  under  the  assump- 
tion of  compensation  uniformly  distributed  and  complete  at  the 
bottom.  Under  a  more  natural  assumption  that  isostatic  compen- 
sation gradually  disappears,  those  heterogeneities  of  density  which 
give  isostatic  compensation  would  gradually  diminish  with  depth 
and  this  diminution  would  extend  to  a  considerably  greater  depth 
than  122  km.  If  heterogeneities  which  act  isostatically  gradually 
disappear,  the  heterogeneities  which  can  be  borne  in  excess  should 
also  be  expected  to  diminish. 

As  to  the  nature  of  the  shell  immediately  below  the  zone  of 
compensation,  Schweydar  has  recently  analyzed  mathematically 
the  results  of  the  measurements  of  earth  tides  by  means  of  the 
horizontal  pendulum. 

The  calculations  were  designed  to  test  the  presence  or  absence 
of  a  viscous  zone  between  an  elastic  crust  and  elastic  interior.  It 
is  concluded  that  even  a  magma  bed  with  a  viscosity  as  high  as 
that  of  sealing  wax  at  house  temperatures  and  a  thickness  of  but 
100  km.  cannot  be  present.  The  assumption  in  best  agreement 


448  JOSEPH  BARRELL 

with  observations  is  that  of  the  presence  of  a  layer  about  600  km. 
thick,  slightly  ductile  (coefficient  io13  to  io14),  existing  beneath 
an  outer  crust  120  km.  thick.1 

By  postulating  such  a  thick  zone  for  isostatic  flow,  the  viscous 
resistances  are  reduced  and  solid  flow  is  made  easier.  It  also  is  in 
conformity  with  the  probability  of  a  gradual  change  of  physical 
state  from  the  rigidity  above  into  a  less  rigid  and  less  stable  tract 
and  this  in  turn  into  a  more  rigid  interior.  Now  if  such  a  thick 
viscous  zone  is  incapable  of  supporting  over  broad  areas  loads 
imposed  by  abnormalities  of  density  above,  it  should  also  be  incap- 
able of  supporting  such  horizontal  inequalities  of  mass  within  it, 
provided  these  are  sufficiently  large.  But  in  order  to  produce  the 
same  gravitative  surface  effects  as  more  superficial  masses,  the 
heterogeneities  of  this  zone  of  viscous  flow  would  in  fact  have  to 
be  much  larger.  A  cylinder  of  the  dimensions  postulated  by 
Gilbert,  if  of  negative  density  and  adjacent  to  another  at  the  same 
depth  but  of  positive  departure  from  the  mean  density,  would  tend 
to  be  underthrust  by  the  latter,  and  the  denser  would  in  turn  tend 
to  be  overflowed  by  the  lighter. 

For  these  reasons  it  is  not  to  be  expected  that  the  same  depar- 
tures from  those  densities  giving  isostatic  equilibrium  which  could 
exist  in  a  rigid  shell  above  would  extend  immediately  below. 

Influence  of  centra spheric  heterogeneity. — To  test  the  question 
of  the  influence  of  heterogeneity  below  the  zone  of  compensation 
a  sphere  will  be  considered.  First,  one  whose  center  is  at  a  depth  of 
319  km.,  0.05  of  the  radius  of  the  earth.  As  a  second  test,  the 
influence  will  be  determined  of  a  sphere  whose  center  is  at  a  depth 
of  637  km.,  o.  io  of  the  earth's  radius.  For  considering  the  attrac- 
tion of  a  mass  at  points  on  the  surface  other  than  at  the  epicenter 
it  is  more  convenient  to  take  the  mass  as  having  the  form  of  a 
sphere  rather  than  a  cylinder,  since  the  mass  of  the  sphere  acts  in 
all  directions  as  if  concentrated  at  its  center.  This  favors,  further- 
more, the  accentuation  of  the  effects  upon  the  surface  over  what 
they  would  be  if  the  outstanding  mass  had  a  stratiform  extension. 

1Dr.  Wilhelm  Schweydar,  "  Untersuchungen  iiber  die  Gezeiten  der  fasten  Erde 
und  die  hypothetische  Magmaschicht,"  V erdjffentlichung  des  k.k.  Preusz.  geodat.  Insti- 
tutes, Neue  Folge  No.  54,  Leipzig,  1912  (B.  G.  Teubner). 


THE  STRENGTH  OF  THE  EARTH'S  CRUST       449 

The  sphere  having  the  same  volume  as  a  cylinder  122  km.  in 
radius  and  122  km.  in  depth  will  have  a  radius  of  in  km.  Let 
a  radius  of  100  km.  and  a  density  of  ±0.025  be  assumed  as  the 
dimensions  and  mass  of  a  standard  sphere  in  this  deep  zone.  If 
the  center  of  such  a  sphere  is  at  a  depth  of  183  km.,  the  same  depth 
as  the  center  of  Gilbert's  postulated  cylinder,  the  anomaly  at  the 
epicenter  will  be  0.021  dyne,  whereas  the  cylinder  gave  an  anomaly 
of  0.023  dyne.  They  are  therefore  nearly  equal  in  effect.  If  the 
center  of  the  sphere  is  placed  at  a  depth  of  319  km.,  making  the 
top  at  2 19  km.,  the  anomaly  at  the  epicenter  becomes  0.0068 
dyne.  Consequently  the  variation  in  density  or  volume  of  the 
sphere  would  have  to  become  three  times  as  great  in  order  that  its 
maximum  anomaly  should  equal  the  mean  observed  anomaly. 
But  as  the  average  anomaly  is  not  measured  at  the  epicenter,  and 
the  maximum  anomalies,  occurring  at  the  epicenters,  are  several 
times  the  observed  mean  anomalies,  this  figure  would  have  to  be 
still  further  multiplied.  To  account,  therefore,  for  the  magnitude 
of  surface  anomalies,  the  disturbing  spheres,  if  with  centers  at  a 
depth  of  319  km.  and  if  of  100  km.  radius,  would  have  to  have 
abnormalities  of  densities  ranging  up  to  0.25  in  order  of  magni- 
tude. If  the  centers  of  the  spheres  were  at  twice  this  depth  the 
abnormalities,  to  produce  the  same  effect,  would  have  to  be  four 
times  as  great  in  mass.  In  a  region  of  which  there  is  no  precise 
knowledge  such  variations  of  density  might  well  occur.  The 
problem  must  therefore  be  investigated  by  means  of  the  gradients 
which  would  result  in  the  gravity  anomalies  and  deflection  residuals 
and  a  comparison  of  these  with  the  gradients  actually  observed 
and  plotted. 

Distribution  of  surface  forces  for  centro spheric  spheres. — For 
masses  as  deep  as  these  the  curvature  of  the  earth  becomes  of 
importance,  but  the  complications  which  it  introduces  into  the 
analytic  treatment  have  been  avoided  by  means  of  a  graphic 
solution. 

In  Fig.  8A,  the  anomaly  is  calculated  for  the  epicenter.  Then 
the  gravitative  force  at  any  other  point  on  the  surface,  such  as 
that  having  a  dip  angle  6,  can  be  determined  by  squaring  the  inverse 
ratio  of  distance.  Multiplying  the  force  at  the  epicenter  by  this 


JOSEPH  BARRELL 


FIG.  8. — Curves  showing  values  for  the  gravity  anomalies  Fv,  and  deflection 
residuals  F/f,  on  the  surface  of  the  earth,  for  spherical  masses  situated  at  depths  of  319 
and  637  km.,  respectively.  Ordinates  measured  at  right  angles  to  earth's  surface. 


TEE  STRENGTH  OF  THE  EARTH'S  CRUST  451 

factor  gives  the  force  at  the  second  point  acting  in  the  direction  of 
the  radius  of  the  attracting  sphere.  This  value  is  laid  off  as  Fr 
and  then  resolved  into  two  components  Fv  and  Fh,  vertical  and 
parallel  respectively  to  the  surface  of  the  earth.  The  ratio  of  Fv 
to  Fh  is  tan  6.  With  increasing  distance  from  the  epicenter  6 
becomes  increasingly  greater  than  it  would  be  if  the  earth's  surface 
were  regarded  as  a  plane.  Therefore  for  distances  of  5  to  10 
degrees  and  more  from  the  epicenter  Fv  begins  to  hold  an  appre- 
ciably higher  ratio  over  Fh  than  it  would  if  curvature  were  neglected. 
It  is  seen  that  Fv=Fh  for  #=45°.  Nearer  the  epicenter  Fv  is  in 
excess;  at  greater  distances  Fh  is  the  greater.  Fv  is  a  maximum 
for  0  =  90°.  Fh  is  a  maximum  for  0=  55°  if  curvature  be  neglected. 
For  the  earth's  curvature  and  a  depth  of  319  km.  to  the  center  of 
mass,  6  is  a  maximum  for  53°=*=.  The  point  giving  this  is  at  a 
distance  from  the  epicenter  of  o.  75  the  depth.  The  ratio  of  maxi- 
mum Fv  divided  by  maximum  Fh  is  approximately  2.7.  In  Fig. 
8C  are  shown  the  effects  of  two  spheres  of  opposite  sign  but  of 
equal  mass.  If  these  two  spheres  were  superposed  they  would 
of  course  completely  neutralize  each  other.  Upon  moving  them 
horizontally  apart  to  i .  5  times  the  depth,  the  maximum  value  of 
Fh  becomes  twice  the  value  for  a  single  sphere.  This  occurs  half- 
way between  them,  and  the  value  of  Fv  for  this  point  is  zero.  The 
ratio  of  maximum  Fv  over  maximum  Fh  becomes  i .  i .  Two 
equal  masses  of  like  sign  would,  on  the  contrary,  give  a  maximum 
value  of  Fv  and  a  zero  value  of  Fh  at  a  point  halfway  between  them. 
These  represent  the  extreme  departures  from  the  case  of  a  single 
spherical  disturbing  mass.  More  distant  masses  show  less  over- 
lapping of  their  fields  of  force  and  tend  to  have  their  individual 
effect  upon  a  point  between  them  neutralized  by  the  larger  number 
of  masses  acting  from  various  directions.  The  values  of  Fv  are 
much  more  under  the  control  of  the  individual  masses  than  are  the 
values  of  Fh. 

Returning  to  the  single  dominating  mass  of  spherical  form  as 
shown  in  Fig.  8A,  let  the  values  of  Fv  and  Fh  be  represented  by 
ordinates  as  shown  in  the  figure;  then  the  surface  representing 
the  gravity  anomalies,  Fv,  would  be  a  dome  of  double  curvature, 
like  a  craterless  volcano;  the  surface  representing  the  deflection 


452  JOSEPH  BARRELL 

force,  Fh,  would  be  in  the  form  of  a  caldera  or  crater  ring.  Accord- 
ing as  the  attracting  mass  departed  in  its  nature  from  a  sphere 
these  shapes  of  the  force  surfaces  would  be  modified,  but  the  general 
character  of  volcanic  cone  and  caldera  would  remain.  Negative 
masses  would  have  the  forms  reversed.  If  the  disturbing  masses 
are  at  distances  apart  which  average  several  times  their  depth, 
then  a  relief  map  of  the  resultant  forces  would  resemble  a  volcanic 
field  with  the  volcanoes  isolated  from  each  other.  If  the  centers 
are  much  closer,  the  relief  map  would  come  more  to  resemble  those 
lunar  craters  which  show  all  degrees  of  superposition  upon  older 
craters.  The  deeper  the  masses  the  broader  the  volcano-like 
curves  of  forces  upon  the  surface,  but  the  lower  will  be  the  relief, 
unless  the  disturbing  spheres  increase  in  mass  with  the  square  of 
the  depth.  Stratiform-like  masses,  such  as  oblate  spheroids  or 
cylindrical  disks,  will  show  less  pronounced  effects  than  the  equiva- 
lent spheres  and  will  simulate  somewhat  the  effects  of  spheres  at 
a  greater  depth.  The  attempt  to  apply  these  principles  as  criteria 
to  the  published  data  must  be  deferred  until  the  influence  of 
abnormal  masses  in  the  zone  of  compensation  has  been  considered 
and  also  in  somewhat  more  detail  the  influence  of  masses  in  other 
forms  than  spheres. 

Influence  of  spheres  within  the  zone  of  compensation. — The  forces 
produced  by  spheres  within  this  zone  are  more  readily  treated 
analytically,  since  it  will  be  seen  that  the  curvature  of  the  earth 
may  be  neglected.  Otherwise  this  topic  is  to  a  considerable  degree 
an  extension  of  the  last.  The  unit  mass  which  it  is  convenient 
to  adopt  for  this  discussion  is  that  of  a  sphere  whose  radius  is  50  km. 
and  density  o.  100,  one-half  the  mass  of  the  sphere  previously  con- 
sidered. Its  center  is  taken  at  a  depth  of  64km.,  o.oi  of  the 
earth's  radius,  and  approximately  at  the  middle  of  the  zone  of 
compensation  as  given  by  Solution  H.  This  gives  the  greatest 
abnormality  of  mass  in  the  middle  of  the  zone  of  compensation 
and  will  approximate  to  the  mean  effect  of  an  outstanding  density 
distributed  uniformly  throughout  that  zone.  At  the  epicenter  the 
attraction  is  wholly  effective  in  producing  gravity  anomaly  and  is 
measured  by  the  formula 

&($*•#) 
D2 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  453 

In  this  d= density,  c  =  constant  of  gravitation,  R  =  radius,  D  = 
depth.  Solving  this  equation  for  the  values  chosen  gives 

F=  .0853  dyne 

Take  the  earth's  surface  as  a  plane  and  any  point  on  it  as  located 
by  the  dip  angle  6,  made  by  a  line  from  the  point  to  the  center  of 
the  sphere  of  outstanding  mass.  Then  the  vertical  component 
Fv  and  the  horizontal  component  Fh  are  given  by  the  following 

equations : 

Fv  (dynes)  =  o .  0853  sin3  0 

Fh  (dynes)  =  0.0853  sin2  0  cos  0 

To  convert  Fh  into  seconds  of  arc  divide  by  0.00475  and 
Fh  (seconds)  = 17 . 94  sin2  0  cos  0 

The  maximum  value  of  Fh  occurs  for  0=  55°  and  is  o. 0328  dyne 
or  6.9  seconds.  The  curves  for  Fv  and  Fh  are  shown  in  the 
unbroken  lines  of  Figs.  9  and  10.  They  are  seen  to  be  quite  close 
in  character  to  the  curves  of  Fig.  8.  Changes  in  the  mass  or  depth 
of  the  sphere  will  serve  to  change  only  the  scales  of  forces  and 
distances  so  that  these  curves  may  be  adapted  readily  to  apply  to 
all  spherical  masses  situated  within  the  lithosphere. 

Influence  of  sum  of  intersecting  spheres  approximately  equivalent 
to  spheroids. — The  analysis  of  the  gravitative  forces  which  a  sphere 
exerts  upon  points  in  an  external  plane  serves  as  a  starting-point 
for  the  consideration  of  the  problem  of  the  influence  of  those  unit 
masses  of  excess  or  defect  of  density  which  exist  in  the  crust.  As 
a  further  step,  any  one  mass  may  be  considered  as  approximating 
in  form  either  to  some  oblate  or  prolate  spheroid  or  to  some  ellip- 
soid of  three  unequal  axes.  But  the  equations  for  the  forces  exerted 
by  spheroids  upon  an  external  plane  are  complicated  and  laborious 
to  solve.  A  sufficient  approximation  to  the  influence  of  a  spheroid 
may  be  made,  however,  by  employing  several  intersecting  spheres 
which  together  give  an  approximation  to  the  right  quantity  and 
distribution  of  mass.  The  influence  of  the  composite  mass  is 
readily  attained  by  summing  up  the  curves  given  by  the  modifica- 
tions for  the  several  spheres. 

In  Fig.  9  the  unbroken  lines,  as  previously  noted,  are  the  curves 
of  force  due  to  the  single  unit  sphere.  The  broken  lines  show  the 


454 


JOSEPH  BARRELL 


'  CASES  B  AND  C 


FIG.  9. — Various  cases  of  horizontal  and  vertical  components  of  the  gravitative 
force  due  to  a  unit  mass  concentrated  into  a  single  sphere  and  expanded  into  several 
intersecting  spheres.  Cases  A  and  B,  three  spheres  in  line.  Case  C,  five  spheres  made 
by  combining  A  and  B  and  omitting  one  central  sphere. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST      455 

curves  of  force  due  to  the  same  unit  mass  expanded  into  three 
spheres  of  the  original  dimensions  and  with  centers  50  kin.  apart 
on  a  horizontal  line.  One-third  of  the  mass  is  therefore  in  each 
sphere  and  the  density  of  each  is  0.033.  In  Case  A  the  line  join- 
ing the  centers  is  at  right  angles  to  the  vertical  section  plane. 
In  Case  B  the  line  joining  the  centers  lies  in  the  section  plane. 
The  dotted  lines  show  Case  C.  In  this  the  unit  mass  is  expanded 
into  five  spheres  of  the  original  size,  each  sphere  possessing,  there- 
fore, one-fifth  of  a  unit  mass,  and  consequently  a  density  of  0.020. 
In  this  case  the  five  centers  are  arranged  in  a  horizontal  plane,  the 
four  outer  spheres  having  their  centers  50  km.  from  the  center  of 
the  inner  sphere. 

The  single  sphere  has  a  volume  given  by  the  formula  V=-irR3, 

o 

in  which  R  =50 km.,  and  a  density  of  o.ioo.  The  three  spheres 
have  a  volume  therefore  of  4wR3  and  a  density  of  0.333.  The 
three  spheres  make  a  solid  of  revolution  whose  semipolar  axis  is 
equal  to  2R,  equatorial  radius  equal  to  R.  Upon  comparing  this 
aggregate  to  a  spheroid  it  is  seen  that  the  double  density  0.667  °f 
the  intersecting  portions  compensates  roughly  for  the  two  re-entrant 
zones  on  each  side  of  the  equator.  To  what  regular  spheroid  does 
it  approximate  in  its  proportions  ?  Let  E  be  the  equatorial  radius 
of  the  spheroid  and  2E  the  semipolar  axis.  The  volume  will  be 

o 

-7J-E3,  equal  to  the  three  spheres  whose  volume  is  4irR3,  or  a  single 

o 

sphere  of  radius  1.44^.  Solving  gives  £=1.14^,  2^=2. 2&R. 
The  spheroid  with  these  semiaxes  is  shown  in  broken  lines  in  Fig.  9. 
This,  then,  is  a  spheroid  which,  if  the  density  be  taken  as  o.  033,  is 
of  exactly  the  same  mass  as  the  original  unit  sphere,  or  the  three 
intersecting  spheres,  and  which  approximates  in  distribution  of 
mass  and  in  gravitative  effect  to  these  three  spheres  as  shown  in 
Fig.  9.  The  nature  of  the  differences  will  be  discussed  later. 

Case  C  shows  five  spheres  of  unit  volume  and  of  density  0.020 
whose  intersecting  portions  would  consequently  have  densities  of 
o .  040  and  o .  060.  In  comparing  the  compound  mass  to  an  oblate 
spheroid  these  intersecting  portions  compensate  roughly  for  the 
re-entrants  between  the  spheres.  The  limiting  dimensions  of  the 


456 


JOSEPH  BARRELL 


whole  in  the  directions  of  the  three  principal  axes  are  R  and  2R. 
Let  it  be  required  to  find  the  value  of  the  equatorial  radius  iP 
and  semipolar  axis  P  of  the  oblate  spheroid  of  equal  volume  in 
which  the  axes  have  these  proportions.  Then 


rv 

i  \ 

I  » 

I  \ 

I    \ 

I     \ 

I     \ 

I      \ 

I       \ 

I        \ 

I  \ 
\ 
\ 


3  3 

P=i.o8R 

It  is  seen  that  the  five  spheres  of  density  o .  020 
have  the  same  mass  as  a  sphere  whose  radius  is 
i.jiR  and  density  0.020;  the  same  mass  also  as 
the  unit  sphere  of  density  o.  100  and  radius  R  and 
the  oblate  spheroid  of  density  o .  020  and  semipolar 
axis  i .  oSR,  equatorial  radius  2.i6R.  The  vertical 
section  of  this  spheriod  is  shown  in  dotted  outline 
in  Fig.  9.  The  distribution  of  mass  and  gravita- 
tive  effect  of  the  five  spheres  will  be 
nearly  the  same  as  for  such  a  spheroid. 
The  nature  of  the  differences,  as  in  cases 
A  and  B ,  will  be 
discussed  later. 
The  effect  of 
the  distribution 
of  mass  along  a 
horizontal  line 
and  in  a  plane 
has  4}een  con- 
sidered in  cases 
A,  B,C.  There 
remains  to  be 
considered  the 
effect  of  the  dis- 
tribution along 


FIG.  10. — Horizontal  and  vertical  components  of  the  gravi- 
tative  force  due,  first,  to  a  sphere  of  32  km.  radius,  density 
o.  100,  depth  to  center  64  km.,  and  second,  to  the  same  mass 
expanded  into  three  such  spheres  with  centers  on  a  vertical 
line  at  depths  of  32,  64,  and  96  km. 


a  vertical  line  and  in  a  vertical  plane.  Case  D,  Fig.  10,  is  given  to 
show  the  effect  of  the  distribution  along  a  vertical  line.  The 
unbroken-line  curves  show  the  values  of  Fv  and  Fh  for  a  sphere 
with  density  and  depth  corresponding  to  the  unit  sphere  previously 


THE  STRENGTH  OF  THE  EARTH'S  CRUST 


457 


used,  but  with  radius  of  32  km.  instead  of  50  km.  The  mass  is 
consequently  but  26  per  cent  of  that  of  the  unit  sphere.  Let  this 
be  expanded  into  three  intersecting  spheres  with  centers  32  km. 
apart  and  arranged  in  a  vertical  line.  The  composite  mass  will 
then  extend  from  the  surface  to  a  depth  of  128  km.  and  correspond 
closely  in  effect  to  a  prolate  spheroid  with  vertical  axis. 

The  resulting  values  of  the  components  of  the  gravitative  force 
for  these  four  cases  which  are  of  importance  for  the  development 
of  criteria  may  be  tabulated  as  follows : 

TABLE  XXV 


UNIT  MASS 

RATIO  OF 

MAXIMUM  Fv 
DIVIDED  BY 
MAXIMUM  Fh 

VALUE  OF  9 

FOR 

MAXIMUM  Fh 

Maximum  Fv 

Maximum  Fh 

Sphere  
Case  A  
Case  B  
Case  C  

Sphere  
CaseD  

0.085 
0.056 
0.056 
0.050 

0-033 
0.024 
0.023 
0.022 

2.6 

2-3 
2.4 

2-4 

2.6 
2.8 

55° 
Si* 

38=*= 
4i=fc 

55 
67-5 

0.26  UNIT  MASS 

Maximum  Fv 

Maximum  Fh 

0.023 
0.041 

O.OOQ 
0.015 

To  complete  the  series  the  curves  should  be  drawn  for  five  inter- 
secting spheres  arranged  in  a  vertical  plane  analogous  to  Case  V, 
first  parallel  and  then  at  right  angles  to  the  plane  of  the  section, 
making  cases  E  and  F,  but  the  general  character  of  the  resulting 
curves  may  be  inferred  from  the  cases  already  given.  Therefore, 
in  order  to  abbreviate  the  discussion,  an  additional  figure  for  cases 
E  and  F  has  been  omitted. 

It  is  seen  from  inspection  of  Figs.  9  and  10  and  Table  XXV  that 
even  for  a  constant  mass  and  center  of  gravity  the  values  of  Fv 
and  Fh  change  rapidly  with  the  changing  form  of  the  mass.  Upon 
the  linear  extension  of  a  sphere  into  a  form  such  as  cases  A  and  B 
the  maximum  value  of  Fv  falls  to  two- thirds  of  its  original  value. 
For  Case  C  it  is  still  less.  The  ratio  of  the  maximum  value  of 
Fv  divided  by  the  maximum  value  of  Fh  is  also  seen  to  change,  but 


458  JOSEPH  BARRELL 

more  slowly,  decreasing  for  the  spreading-out  of  the  mass  in  a 
horizontal  direction,  increasing  for  a  linear  vertical  extension.  For 
the  spheroids  whose  axes  are  in  the  relation  of  i  to  2  and  to  which 
the  spheres  are  equivalent  in  volume,  the  changes  would  be  still 
more  marked.  This  is  .because  the  duplications  of  mass  due  to  the 
intersecting  spheres  are  near  the  center  and  in  Case  C  a  top  view 
would  show  considerable  deficiencies  of  mass  in  the  equatorial  zone 
between  the  spheres  and  the  spheroid.  These  intersecting  spheres 
are  consequently  more  effective  in  producing  gravity  anomaly 
and  somewhat  more  effective  in  producing  deflection  of  the  vertical 
in  the  zone  of  maximum  deflection.  For  cylindrical  disks  of  the 
same  mass  and  proportions  as  the  spheroids  the  changes  away 
from  the  values  for  a  sphere  would  be  still  greater,  since  a  greater 
proportion  of  the  mass  would  be  removed  from  the  center  to  the 
edges  of  the  body. 

Distinctive  effects  of  individual  spheres  and  spheroids. — In  the 
subjection  of  deflection  measurements  to  the  hypothesis  of  isostasy 
Hayford  found  it  necessary  to  consider  the  effects  of  topography 
to  a  distance  of  4,125  km.  and  in  the  determination  of  gravity 
anomalies  the  topography  of  the  whole  earth  and  its  compensation 
were  considered.  To  what  extent  then  do  distant  masses  affect 
the  local  residuals  and  vitiate  any  attempt  to  analyze  the  effects 
of  local  masses  ?  In  answer,  it  is  seen  that  the  effects  of  distant 
masses  are  negligible.  It  is  the  great  topographic  contrast  of 
ocean  basins  and  continental  platforms,  to  a  lesser  extent  the  large 
variations  of  relief  within  these  areas,  which  require  their  effects 
to  be  considered  to  such  a  great  distance.  But  it  is  found  that 
this  larger  relief  of  the  crust  is  nine-tenths  compensated  and  the 
outstanding  masses  so  far  as  known  do  not  show  any  marked 
segregation  as  to  sign  within  the  continental  areas  as  opposed  to 
the  oceanic  areas.  Furthermore,  the  unit  areas  departing  markedly 
from  isostatic  equilibrium  are  much  smaller  than  these  major 
segments  of  the  crust.  Therefore  where  it  is  the  effect  of  the  out- 
standing masses  which  is  under  consideration  they  are  seen  to 
be  individually  small  in  comparison  with  the  greater  relief  features 
of  the  globe,  and  further,  they  mutually  cancel  their  effects.  The 
local  outstanding  masses  are  furthermore  of  the  same  order  of 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  459 

magnitude  as  the  more  distant  ones  and  therefore  the  effects  of 
the  distant  masses  sink  to  negligible  quantities  in  comparison,  in 
accordance  with  the  law  of  inverse  squares.  The  general  agree- 
ment in  the  magnitude  of  the  departures  from  isostasy,  as  shown 
by  deflection  residuals  and  gravity  anomalies,  shows,  furthermore, 
that  deep  nucleal  heterogeneities  can  have  no  large  and  broad 
regional  effects,  since  such  would  affect  the  gravity  measurements 
within  a  broad  central  circle  to  a  greater  degree  than  they  would 
the  deflections  of  the  vertical.  Therefore  each  region  is  seen  to 
offer  its  local  problems  and  the  dominating  centers  of  outstanding 
masses  may  be  readily  determined  save  where  several  such  masses 
are  contiguous,  especially  if  of  opposite  sign.  Let  attention  be  given 
then  to  those  features  in  the  influence  of  outstanding  masses  which 
are  indicative  of  the  form  and  depth  of  the  attracting  mass. 

Let  the  depth  to  the  center  of  mass  be  D.  Then  it  is  seen  that 
for  ellipsoids  near  the  surface  in  which  the  major  axes  are  twice 
the  minor  axes  the  influence  of  form  has  mostly  disappeared  at 
horizontal  distances  from  the  epicenter  or  from  2D  to  $D,  and  at 
greater  distances  the  curves  become  practically  coincident.  For 
greater  departures  from  the  spherical  form  the  distances  before  the 
curves  approach  those  given  by  a  spherical  mass  are  still  greater. 
At  these  distances  where  the  curves  approach  those  of  spheres, 
the  effects  of  the  form  of  the  mass  could  not  be  distinguished,  but 
the  curves  are  so  flat  that  neither  could  the  effects  of  depth  be 
readily  evaluated.  For  instance,  the  curves  of  force  at  ^D  to  6D 
for  a  sphere  of  mass  M  at  depth  D  would  be  approximately  the 
same  as  for  a  sphere  of  mass  ^M  at  a  depth  of  2D.  Furthermore, 
at  these  distances  from  the  epicenter  the  forces  are  so  small  in 
proportion  to  the  maxima  for  the  same  mass  that  other  outstanding 
masses  would  greatly  change  their  value  and  prevent  a  correct 
analysis.  Therefore,  to  be  determinative,  observations  must  be 
made  at  a  number  of  points  between  the  epicenter  E  and  a  distance 
not  more  than  double  that  which  at  the  point  M  gives  the  maximum 
value  to  Fh.  It  is  seen  that  if  a  mass  symmetrical  about  a  vertical 
axis  departs  widely  from  the  spherical  form,  this  will  be  detected 
by  noting  the  ratio  of  maximum  Fv  to  maximum  Fh,  the  latter 
being  measured  along  any  line  radiating  from  the  epicenter.  If 


460  JOSEPH  BARRELL 

the  mass  is  unsymmetrical  about  a  vertical  axis,  observations  must 
be  made  along  at  least  two  lines  at  right  angles  to  each  other. 

A  minimum  number  of  observations  will  define  an  isolated 
outstanding  mass,  but  if  several  masses  have  their  fields  of  force 
notably  overlapping,  a  larger  number  of  observations  becomes 
necessary  in  order  to  differentiate  their  effects.  An  inspection  of 
Figs.  9  and  10  shows  that  the  shape  of  the  curve  of  Fv  between  the 
epicenter  and  a  distance  where  it  falls  to  one-tenth  the  maximum 
value  has  more  distinctive  relation  to  the  form  of  the  immediately 
adjacent  mass  than  has  the  curve  for  Fh.  But  the  available 
geodetic  data  supply  less  information  regarding  the  gradients  of 
the  gravity  anomalies  than  for  the  deflection  residuals.  The  latter 
are  given  along  a  certain  belt  of  triangulation  stations,  whereas 
the  gravity  stations  are  located  at  long  distances  apart.  Further- 
more, but  few  of  the  gravity  stations  coincide  with  deflection 
stations.  The  present  analysis  will  therefore  rest  upon  the  data 
giving  the  curve  for  Fh.  This  curve  is  flat  at  the  top,  so  that  the 
data  will  readily  give  the  approximate  value  of  the  maximum 
but  will  not  determine  closely  its  distance  from  the  epicenter.  The 
value  of  6  will,  however,  be  determined  ordinarily  on  two  sides  of 
the  epicenter  by  means  of  the  deflection  residuals  and  the  mean 
will  give  a  more  reliable  figure  than  either  alone.  But  according 
to  the  form  of  the  mass  within  those  limits  shown  in  Figs.  9  and 
10  the  value  of  6  may  range  from  38°  to  67!°.  If  the  abnormal  mass 
is  assumed  to  have  a  spherical  form,  its  center  will  lie  at  an  angle 
of  55°  below  the  maximum  value  of  Fh  and  at  a  depth  1.4  the 
distance  to  the  epicenter.  The  error  in  locating  the  points  of  epi- 
center and  maximum  Fh  may  cause  the  estimate  of  depth  to  be  in 
error  20  per  cent  and  yet  this  figure  will  show  definitely  whether 
the  sphere  lies  within  the  zone  of  compensation  or  in  the  centro- 
sphere.  If  the  mass,  however,  is  in  reality  a  horizontally  elongate 
mass,  the  change  in  the  distance  EM  from  the  epicenter  to  the  point 
of  maximum  Fh  in  two  directions  at  right  angles  to  the  epicenter 
will  show  that  fact.  A  check  on  the  form  of  the  mass  may  be 
obtained  if  the  value  of  the  deflection  curve  is  known  with  fair 
accuracy  to  a  distance  from  the  epicenter  of  three  times  the  distance 
of  the  maximum.  Let  the  distance  to  the  point  of  maximum  value 


THE  STRENGTH  OF  THE  EARTH'S  CRUST       461 

be  M.  The  ratios  of  the  value  of  Fh  at  M  to  the  value  at  $M 
are  given  below  as  measured  from  the  curves. 

RATIOS  OF  Fh  AT  M  TO  FA  AT  3  M 

Sphere 2.4 

Case  A 2.1 

CaseB .  ..." 4-3 

Case  C 3-4 

Case  D 2.0 

It  is  seen  that,  from  the  difficulty  in  the  precise  location  of  M  and 
hence  the  difficulty  in  locating  a  point  as  at  iM  or  3^,  and  further- 
more the  probability  that  other  masses  may  influence  to  a  degree 
not  readily  determinable  the  value  at  $M,  this  test  cannot  ordinarily 
be  determinative.  However,  for  spheroids  with  polar  axes  vertical, 
markedly  prolate  masses  give  a  ratio  distinctly  smaller  than  for  a 
sphere,  the  curve  of  deflections  falling  off  more  abruptly;  markedly 
oblate  masses,  on  the  other  hand,  give  a  ratio  distinctly  larger  than 
that  for  a  sphere,  the  curve  of  deflection  beyond  the  point  of 
maximum  being  flatter. 

If  the  outstanding  mass  approximates  to  an  oblate  spheroid  with 
polar  axis  vertical,  then  the  assumption  of  a  spherical  nature  will 
locate  the  center  of  mass  too  deep  and  imply  a  greater  mass  than 
really  exists.  If,  on  the  contrary,  the  form  of  the  outstanding 
mass  approaches  the  form  of  a  vertical  prolate  spheroid,  the  inter- 
pretation of  the  deflections  as  caused  by  a  sphere  would  locate  the 
center  of  figure  too  high  and  give  it  too  small  a  mass. 

Suppose  the  curves  for  Fh  shown  in  cases  C  and  D  have  their 
maxima  well  determined  in  position  and  in  magnitude,  but  that 
the  values  of  the  curves  at  distances  two  or  three  times  beyond  are 
not  accurately  known.  Let  these  maxima  be  interpreted  as  pro- 
duced by  spheres.  The  depth  and  masses  of  the  spheres  which  give 
these  maximum  deflections  will  be  too  great  by  the  amounts  shown 
in  the  following  tabulation  (Table  XXVI,  p.  462). 

Where  the  data  are  sufficiently  complete  the  form  and  depth 
of  mass  may  both  be  determined,  though  a  high  precision  is  not 
to  be  expected.  But  in  most  cases  with  the  present  geodetic  data 
the  form  of  the  mass  will  not  be  determinable  and  all  that  can  be 


462 


JOSEPH  BARRELL 


done  is  to  interpret  the  deflections  as  produced  by  spherical  masses. 
What  then  are  the  geological  suggestions  as  to  whether  vertical 
prolate  or  oblate  forms  may  be  expected  to  characterize  the  larger 
outstanding  masses?  In  the  one  case  the  error  of  interpretation 
will  be  to  make  the  masses  appear  too  small  and  shallow;  in  the 
other  case,  to  make  them  appear  too  great  and  deep. 

TABLE  XXVI 

ERRORS  DUE  TO  INTERPRETATION  AS  SPHERES  OF  UNIT  MASSES  AT  DEPTH  D 


FORM 

DATA 

ASSUMPTION 

RESULTING 
INTERPRETATION 

Fh  Max. 

EM. 

0 

Depth  to 
Center 

Mass 

Case  C  (True)  

4?5 
4-5 
3-05 
3-05 

74km. 
74km. 
27  km. 

27  km. 

41° 
55 
67-5 
55 

i.oD 
i.6D 
i.oD 
o.6D 

i.oM 
i.8M 
i.oM 
o.6M 

Interpretation  as  a  sphere  .... 
Case  D  (True)  
Interpretation  as  a  sphere  .... 

Stocks,  and  especially  volcanic  pipes,  approach  in  form  to  vertical 
cylinders,  but  these  are  merely  connecting  structures.  On  the 
other  hand,  mountain  ranges  and  geosynclines,  although  linearly 
extended,  are  of  breadth  which  is  great  in  comparison  with  the 
depth  of  excess  or  defect  of  density.  Laccoliths  and  regional 
extrusions  are  also  broad  in  comparison  with  depth.  The  relations 
as  regards  the  great  intrusive  masses  are  not  so  clear,  but  erosion 
exposes  batholiths  over  progressively  greater  areas;  and  whole 
provinces  which  exhibit  regional  metamorphism  give  suggestions 
that  they  are  underlain  by  widespread  igneous  bodies.  The 
hydrostatics  of  the  magmas  and  their  differentiation  into  masses 
of  unlike  density  would  also  give  tendencies  to  layers  and  horizontal 
extensions  of  the  larger  masses  of  abnormal  density.  These  would 
depart,  then,  from  the  form  of  spherical  masses  in  the  direction  of 
oblate  spheroids  with  their  equators  in  a  horizontal  plane.  Nar- 
rower belts  of  disturbance  like  that  which  passes  through  Washing- 
ton, D.C.,  may,  on  the  other  hand,  tend  to  have  the  form  of  vertical 
plates.  Therefore  in  none  but  the  smaller  and  connecting  struc- 
tures are  there  geological  suggestions  of  vertical  prolate  form. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  463 

The  summation  of  this  discussion  shows  that  if  in  the  first 
assumption  as  to  the  form  of  abnormal  masses  they  be  taken  as 
spheres,  then  the  determination  of  the  depth  of  the  center  of  mass 
by  means  of  the  curve  for  Fh  and  the  location  of  the  point  of  maxi- 
mum value  is  more  likely  to  overestimate  than  underestimate 
the  depths  and  masses.  As  the  object  of  this  investigation  is 
especially  to  find  the  depth  of  masses  and  to  test  the  hypothesis  of 
centrospheric  heterogeneity  as  a  cause  of  deflection  residuals  and 
gravity  anomalies,  it  is  desirable  to  have  the  error  of  interpreta- 
tion in  the  direction  of  indicating  a  depth  too  great  rather  than 
too  small.  Therefore  the  initial  assumption  that  the  outstanding 
masses  are  spheres  is  justified  by  the  geologic  probabilities  and  is 
found  in  Section  B  to  be  justified  by  the  geodetic  evidence.  The 
next  topic  will  therefore  develop  further  the  subject  of  the  interpre- 
tation as  spheres  with  the  view  to  utilizing  the  geodetic  data. 

Depths  of  spheres  whose  epicenters  are  not  on  the  line  of  traverse. — 
If  the  primary  purpose  of  a  geodetic  investigation  were  the  deter- 
mination of  the  location  of  the  epicenters  of  abnormal  masses  and 
then  the  measurement  of  their  form,  size,  and  depth,  a  series  of 
gravity  and  deflection  measurements  could  be  made  in  a  line  passing 
above  the  mass  and  near  the  epicenter.  The  preceding  discussion 
would  then  directly  apply.  In  only  a  few  localities,  however,  will 
a  line  of  triangulation  stations,  located  in  connection  with  the 
measurement  of  the  earth's  surface,  pass  approximately  over  the 
center  of  a  large  outstanding  mass.  How  then,  from  the  locations 
and  values  of  the  deflection  force  along  any  linear  belt  of  measure- 
ments, shall  the  location  of  the  epicenter  and  depth  to  the  center 
of  an  abnormal  mass  to  one  side  of  the  line  of  traverse  be  deter- 
mined ? 

In  Fig.  1 1  is  developed  a  method  for  the  solution  of  this  problem. 
In  accordance  with  the  preceding  discussion  and  the  reconnaissance 
nature  of  a  first  investigation,  let  it  be  assumed  that  isolated 
abnormal  masses  approach  a  spherical  form;  that  is,  that  a  mass 
may  be  regarded  as  concentrated  at  a  point.  Take  the  center  of 
the  mass  as  the  center  of  co-ordinates  and  the  axis  X-X  as  lying 
parallel  to  the  line  of  traverse.  The  epicenter  is  at  E.  Then  the 
vertical  distance  from  the  center  to  the  epicenter  is  D.  The 


464 


JOSEPH  BARRELL 


Confour  map  for  deflections  in  the  horizontal  plane 

Values  Or  raVio.^   Jt*L    0^ca'e  in  kilomerers  corresponding  toAsphere  at  deorh^akm 

Fig.A 


rejection  of  points  of  maxima  on  to 
rheverrical  plane X-X 


Fx,Def lections  in  fhe  plane  of  rhe  secMons 


F^,De  flections  at  right  angles  ro  plane  of  sections 


FIG.  ii. — Values  of  the  components  of  Fh  on  traverse  lines  which  pass  at  various 
distances  EE'  from  the  epicenter  £  of  a  spherical  mass  at  depth  D;  EE'  being  measured 
in  terms  of  D. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  465 

distance  EE'  from  epicenter  to  the  traverse  line  is  y.  Then  the 
co-ordinates  of  any  point  on  the  surface  are  x,  y,  and  D.  At  any 
such  point  the  component  in  the  horizontal  plane  of  the  gravitative 
force  due  to  the  outstanding  mass  is  Fh.  But  Fh  is  directed  radially 
from  all  points  to  the  epicenter.  Fh,  consequently,  is  subdivided 
into  two  components  at  right  angles.  Let  that  acting  in  the  line 
of  the  section  be  called  Fx,  that  at  right  angles  be  called  Fy,  parallel 
respectively  to  the  X  and  F  axes. 

The  value  of  Fx  for  any  point  is  given  by  the  equation 

3*     ( 


and  the  value  of  Fy  is  given  by  the  equation 

Fy=dc-Tr.D* 
3 


Fig.  ii  shows  the  results  of  the  solution  of  these  equations. 
In  Fig.  nA  is  drawn  a  contour  map  of  the  deflection  force  produced 
by  the  unit  sphere.  The  deflection  force  Fh  at  any  point  is  measured 
by  the  contour  map  and  is  directed  toward  the  epicenter.  The 
lines  of  equal  deflection  are  seen  to  be  circles  with  center  at  E. 
They  show  in  plan  the  values  which  were  shown  in  section  by  the 
full  lines  for  Fh  in  Figs.  9  and  10.  The  contours,  as  previously 
discussed,  are  seen  to  give  the  form  of  a  volcano  whose  crater  has 
a  rounded  rim  and  a  conical  interior  reaching  to  the  epicenter. 

Now  let  a  number  of  parallel  sections  be  taken  at  horizontal 
distances  from  the  epicenter  equal  to  o.  oZ>,  o.  5^,  etc.  The  curves 
for  Fx  for  each  section  are  shown  in  Fig.  nC  and  for  Fy  in  Fig. 
nD.  The  points  of  maximum  value  for  each  section  are  indicated 
in  A,  B,  C,  and  D  and  through  these  points  are  drawn  the  curves 
which  are  loci  of  maxima.  For  Fy  there  is  a  single  maximum  for 
each  section  and  this  is  situated  at  #  =  o.  For  Fx  there  are  in 
each  section  two  equal  maxima  but  of  opposite  sign,  one  for  a 
plus,  the  other  for  an  equal  minus  value  of  x.  In  Fig.  nC  only 
one  side  of  the  curve  is  shown,  that  for  plus  values  of  x. 

Fig.  nB  shows  the  points  giving  maximum  values  of  Fx  pro- 
jected onto  the  vertical  plane  passing  through  X-X.  The  dip 


466  JOSEPH  BARRELL 

angle  8  measures  the  slope  from  the  point  of  maximum  value  to 
the  center  of  mass.  Here  is  shown  its  value  when  projected  onto 
the  plane  X-X.  This  projected  angle  is  seen  to  grow  smaller  with 
each  more  eccentric  position  of  the  section  plane. 

For  section  o.  oD  the  real  value  of  0  is  55° 
For  section  i .  oD  the  projection  of  6'  is  45° 
For  section  2.oD  the  projection  of  9"  is  32° 
For  section  3 .  oD  the  projection  of  6'"  is  24° 

Therefore  it  is  seen  that  if  the  traverse  line  were  assumed  to  pass 
through  the  epicenter  of  all  disturbing  masses,  the  error  introduced 
would  be  to  show  the  center  of  mass  deeper  than  it  really  is.  The 
nature,  however,  of  the  geodetic  data  permits  this  assumption  to 
be  eliminated  and  the  distance  EE'  to  the  section  plane  to  be 
approximately  determined.  An  error  up  to  o.$D  will  not  involve 
much  error  in  the  resultant  depth  as  determined  by  the  projection 
of  0.  At  each  station  along  a  line  of  triangulation1  both  Fx  and 
Fy  are  determined  and  their  resultant  points  toward  the  center  of 
gravitative  control.  Each  station  gives  an  independent  deter- 
mination of  this  resultant  and  the  intersection  of  two  resultants 
if  accurately  determined  and  due  to  the  gravitative  force  of  a  single 
symmetrical  mass  would  give  an  accurate  location  of  the  epicenter, 
measuring  its  distance  and  direction  from  the  traverse  line.  The 
data  in  many  cases  permit  as  many  as  three  or  four  resultants  to 
be  drawn,  the  size  of  the  triangle  of  their  mutual  intersections 
showing  to  what  degree  the  forces  may  be  ascribed  to  a  single 
center.  The  relative  positions  of  the  line  of  section  and  epicenter 
of  mass  are  thus  in  many  cases  approximately  established. 

But  although  the  relative  position  of  epicenters  and  traverse 
line  are  thus  ascertained,  the  depth  of  the  masses  remains  to  be 
solved.  In  Fig.  u  the  distance  of  the  traverse  line  from  the  epi- 
center is  given  in  terms  of  D,  but  this  is  the  unknown.  Two  inde- 
pendent methods  lead  up  to  the  solution  of  D. 

First,  on  any  line  of  section  occurs  a  zero  point  for  Fx.  Let 
this  be  called  E' '.  On  each  side  of  the  zero  point  for  Fx  occurs  a 

1  As  shown  in  illustration  No.  3,  Hayford,  Supplementary  Paper. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  467 

maximum  value  for  Fx.    Let  the  point  of  this  maximum  be  called 
M.    These  two  values  are  given  by  the  geodetic  data.     Then  for 


any  spherical  mass  the  ratio  of  EVT?  increases  with  increase  in  the 

77  T<r 

ratio  of  —~-  but  not  as  a  rectilinear  function.     This  relation  of 

ratios  is  shown  graphically  in  Fig.  nE,  in  which  the  abscissas  are 

EE' 
the  values  of  -yr-  and  the  ordinates  are  the  corresponding  ratios 

EE' 

°f  7vT7-     This  ratio  may  be  determined  from  the  geodetic  data 
H,  M 

but  from  the  location  of  the  maxima,  not  their  amount. 

The  second  method  for  determining  the  depth  depends  upon 
the  ratio  of  the  maximum  value  of  Fy  for  any  traverse  line  to  the 
maximum  value  of  Fx  for  the  same  traverse  line,  thus  being  depend- 
ent upon  the  relative  values  of  the  maxima  and  not  their  location. 


This  ratio  also  increases  with  increase  in  the  ratio  of  -jr-  but  not 

as  a  rectilinear  function.  The  maximum  Fx  for  a  section  at  any 
distance  from  E  is  shown  in  Fig.  nC.  For  example,  if  E'E  = 
o  .  $D  the  maximum  Fx  for  the  unit  sphere  is  5  .  $2"  .  The  maximum 
Fy  is  shown  in  Fig.  nD,  and  for  E'E  =  o.  $D  is  6.42".  The  ratio 
of  5.52  to  6.42  is  1.16.  These  ratios  are  shown  in  Fig.  nF 
for  all  traverse  lines  up  to  a  distance  of  3  .  oD  from  the  epicenter. 
The  value  of  this  ratio  is  given  by  the  geodetic  data  for  any 
traverse  line  and  hence  the  distance  to  the  epicenter  is  given  in 
terms  of  D. 

In  conclusion  on  this  topic  it  may  be  said  that  the  curves  shown 
in  Figs.  nE  and  nF  are  independent  of  the  mass  or  volume  of 
the  sphere,  depending  only  upon  its  depth,  and  are  adapted  to 
use  with  the  geodetic  data.  It  is  seen  from  both  curves  that  the 
significant  ratios  change  in  value  rapidly  with  increasing  distance 
of  the  traverse  line  from  epicenter  up  to  a  distance  E'E  =  D, 
but  beyond  this  point  the  change  in  the  value  of  the  ratios  be- 
comes progressively  small  as  compared  to  a  change  in  the  distance 
of  the  section  plane.  The  method  is  therefore  well  adapted  for 


468  JOSEPH  BARRELL 

determining  the  depth  of  the  outstanding  masses  assumed  as  spheres, 
provided  the  section  line  is  not  farther  from  the  epicenter  than  the 
latter  is  above  the  center  of  mass.  The  method  may  be  used, 
however,  with  less  precision  for  distances  of  the  traverse  line  E'E 
up  to  2D.  Beyond  this  distance,  however,  influences  of  other 
masses  or  errors  in  the  geodetic  data  would  be  likely  to  give  wholly 
erroneous  results,  not  distinguishing  between  a  large  excess  of  mass 
at  a  great  depth  or  a  smaller  one  at  a  much  less  depth. 

[To  be  continued] 


VOLUME  XXII  NUMBER  6 

REPRINTED  FROM 
THE 

JOURNAL    OF   GEOLOGY 

SEPTEMBER-OCTOBER,  1914 


THE  STRENGTH  OF  THE  EARTH'S  CRUST 


JOSEPH  BARRELL 
New  Haven,  Connecticut 


PART  V.       THE  DEPTH  OF  MASSES  PRODUCING  GRAVITY 
ANOMALIES  AND  DEFLECTION  RESIDUALS 

SECTION  B 

APPLICATIONS  OF  CRITERIA  TO  DETERMINE  THE  LIMITS  OF  DEPTH, 

FORM,  AND  MASS 

DEPTHS  INDICATED  BY  THE  MAP  OF  DEFLECTION  RESIDUALS    .      .      .  537 

General  Relations  Shown  by  Deflections 537 

Detailed  Study  of  the  Texas-Kansas  Region          539 

OTHER  INDICATIONS  REGARDING  DEPTH  OF  OUTSTANDING  MASSES        .  547 

Deflections  Caused  by  Linear  or  Dike-like  Masses       ....  547 

Indeterminate  Evidence  from  Anomaly  Contours        ....  548 

RELATION  OF  DEPTH  OF  OUTSTANDING  MASSES  TO  HYPOTHESES  RE- 
GARDING DISTRIBUTION  OF  COMPENSATION  551 

DEPTHS   INDICATED   BY   THE   MAP    OF   DEFLECTION  RESIDUALS 

General  relations  shown  by  deflections. — Hay  ford  gives  a  plate1 
which  shows  all  the  residuals  of  Solution  H.  These  are  laid  off  as 
arrows  and  show  graphically  the  magnitude  and  direction  of  the 
portions  of  the  deflections  which  are  outstanding  after  allowance 
is  made  for  the  deflections  calculated  according  to  Solution  H. 
They  therefore  show  the  excesses  or  deficiencies  of  mass  in  the  crust 

1  Supplementary  Paper,  illustration  No.  3;  also  Bowie,  1912,  illustration  No.  5. 
Vol.  XXII,  No.  6  537 


538  JOSEPH  BARRELL 

as  measured  against  the  demands  of  the  hypotheses  made  in  that 
solution.  In  Fig.  12 A  is  reproduced  a  portion  of  Hayford's  chart. 
A  general  inspection  of  the  map  of  the  residuals  of  Solution  H 
shows  that  many  of  the  large  deflections  of  opposite  sign  lie  compar- 
atively close  together.  On  a  line  connecting  two  stations,  Fx  is  the 
component  of  the  deflection  which  lies  in  that  line,  Fy  is  the  com- 
ponent at  right  angles  to  that  line.  In  most  cases  not  enough 
stations  are  located  on  an  approximately  straight  line  to  permit 
well-defined  curves  to  be  drawn  for  Fx  and  Fy.  But  it  has  been 
shown  that  for  spheres  and  other  concentrated  masses  the  curve 
for  Fx  rises  steeply  from  zero  to  maximum  value  and  sinks  away 
more  gently  beyond.  Even  for  flat  disks  the  outer  part  of  the 
deflection  curve  will  be  flatter  than  for  the  inner  part.  Random 
locations  on  the  curve  are  therefore  more  likely  to  give  the  maximum 
measurement  at  some  point  beyond  the  real  maximum  rather  than 
at  some  point  between  the  epicenter  and  the  real  maximum.  Using 
these  stations  giving  maxima  for  Fx  as  if  they  were  at  the  points  of 
real  maxima  will  therefore  give  on  the  average  too  great  a  distance 
from  the  center  to  the  point  of  real  maximum  Fx  and  consequently 
too  great  a  depth  to  the  centers  of  attraction.  Interpreting  the 
disturbing  masses  as  spheres  is  also  an  assumption  likely  to  give 
too  great  a  depth,  as  is  indicated  later.  Minor  centers  of  out- 
standing mass  will  affect  the  positions  of  the  points  of  maxi- 
mum value,  but  in  a  sufficient  number  of  examples  this  effect  will 
largely  cancel  out.  The  tabulation  of  the  distances  measured  from 
Hayford's  map  between  ten  pairs  of  notable  Fx  maxima  is  given 
in  Table  XXVII. 

It  is  seen  that  the  distance  between  these  maxima  is  more 
largely  dependent  upon  the  length  of  the  sides  of  the  geodetic 
triangles  than  upon  the  depth  to  center  of  mass,  since  in  less  than 
half  of  these  illustrations  did  a  station  fall  between  the  two  maxima. 
The  distance  between  the  real  maxima  is  then  probably  somewhat 
greater  than  86  km.,  the  average  of  the  six  distances  without 
intervening  stations,  but  is  probably  somewhat  under  the  general 
mean  of  no  km.  This  mean  distance  of  no  km.  between  ten 
notable  maxima  of  Fx  corresponds  to  a  mean  depth  of  spheres  of 
79  km.  Considering  the  various  assumptions  made,  it  is  seen  that 


THE  STRENGTH  OF  THE  EARTH'S  CRUST 


539 


the  mean  depth  of  masses  producing  these  deflections  is  probably 
much  less  than  79  km.  They  belong,  therefore,  to  the  outer  half  of 
the  zone  of  compensation. 

TABLE  XXVII 

DISTANCES  BETWEEN  ADJACENT  LARGE  DEFLECTIONS  OF  OPPOSITE  SIGN 


Locality  of  Mass 

Sign  of  Mass 

Numbers  of 
Stations 

Stations 
between 

Distances 
between  in  Km. 

New  Jersey  

+  ' 

2ZZ—  IA2 

o 

IIO 

Kentucky—  Ohio 

-J- 

8«;-84 

o 

IOO 

Georgia-Florida  
Florida                  

+ 
+ 

292-294 
200—300 

I 
o 

1  60 
14"? 

Michigan-Indiana  
Illinois 

+ 
-f 

344-346 

7^—74 

i 
o 

165 
OO 

Nebraska 

-f 

•2  27—  32Q 

I 

1  7O 

Colorado 

CTQ—  ry 

I 

IOO 

California 

_ 

238-236 

o 

CQ 

California        

_ 

245-^246 

o 

22 

Mean  

IIO 

There  are  other  areas,  however,  as  in  the  Adirondacks,  Maine, 
Michigan,  and  the  Great  Basin,  where  the  distance  between  the 
large  deflections  of  opposite  sign  is  considerably  greater.  So  far 
as  this  relation  goes  they  could  be  due  either  to  broad  outstanding 
masses  in  the  zone  of  compensation  or  to  much  greater  but  more 
concentrated  masses  in  the  nucleus  beneath.  But  the  general 
relations  to  the  magnitude  and  location  of  the  gravity  anomalies 
as  discussed  later  under  that  subject  suggest  that  in  so  far  as  the 
evidence  is  determinative  these  broader  areas  are  also  due  to  broad 
excesses  or  deficiencies  in  the  outer  crust,  not  to  masses  in  the 
centrosphere.  The  data  are  not,  however,  in  all  areas  of  a  suffici- 
ently complete  nature  to  give  determinate  solutions.  In  other 
areas,  however,  detailed  study  following  the  lines  of  criteria  previ- 
ously developed  can  bring  out  very  definite  results  in  regard  to  the 
location  and  depth  of  masses  in  spite  of  the  interference  of  the 
fields  of  force  from  various  centers.  An  example  of  what  may  be 
done  by  a  detailed  examination  is  shown  under  the  next  topic. 

Detailed  study  of  the  Texas-Kansas  region. — Fig.  12 A  shows  the 
deflections  as  given  on  a  north-south  line  of  triangulation  1,000  km. 
in  length.  The  gravity  anomalies  are  shown  for  distances  of  200 
km.  on  each  side  of  the  traverse.  The  stations  are  sufficient  in 


540 


JOSEPH  BARRELL 


4' 

•f  n 


-009 


Lonq.96° 


»+.o32     Long.  100° 


DeflecHon 
residuals  and 
gravity  anomalies 


*"-.027 


B 


Ftj  componenhs 
f 


of  deflecHon  residuals  \ 


-5  - 


<>^ 

..  /^Sr 


F,x  cotnponen|-s 

of  deflecKon  residuals 


Scale  for  residuals"5 
o 


Scale  for  distances 
100       o  sookm 

llUlllllll I I I I I 


FIG.  12. — Residuals  of  Solution  H  and  gravity  anomalies  in  Texas,  Oklahoma, 
and  Kansas,  with  the  interpretation  of  the  outstanding  masses  in  terms  of  equivalent 
spheres. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  541 

number  and  sufficiently  close  to  a  straight  line  to  permit  the  applica- 
tion of  the  principles  previously  discussed. 

The  residuals  are  given  in  the  north-south  and  east- west  direc- 
tions. The  broken  lines  give  the  resultants.  Their  convergence 
indicates  that  there  are  two  large  and  controlling  positive  masses 
marked  on  the  map  as  2  and  5.  To  account  for  the  local  variations 
shown  by  the  resultants  from  station  to  station  it  is  necessary, 
however,  to  locate  smaller  masses  of  positive  or  negative  nature 
approximately  as  shown  at  i,  3,  4,  and  6.  There  must  be  of  course 
many  other  centers  of  moderate  disturbance  within  the  area  of 
400,000  km.  which  is  shown,  but  such  as  exist  are  far  enough  from 
the  line  of  section  not  to  exert  an  appreciable  influence.  It  is 
noteworthy  that  gravity  stations  only  200  km.  from  the  line  of  the 
traverse  can  show  anomalies  as  large  as  —0.029,  +0.032,  and 
—0.027  dyne  without  the  masses  which  give  these  anomalies 
showing  appreciable  control  over  the  deflections  on  the  line  of 
traverse.  Their  areas  of  influence  are  therefore  restricted.  The 
limited  influence  of  these  masses  giving  anomalies  somewhat 
above  the  average  and  at  a  moderate  distance,  and  the  small 
masses  locally  modifying  the  deflections  both  serve  to  show  the 
importance  of  nearness  of  location.  This  limitation  of  control 
over  the  deflections,  restricted  to  distances  of  less  than  100  to  200 
km.,  is  itself  an  indication  that  these  outstanding  masses  lie  within 
the  zone  of  compensation,  otherwise  their  effects  would  be  more 
far-reaching. 

The  residuals  permit,  however,  a  much  more  detailed  solution 
to  be  made.  As  a  first  approximation  assume  the  outstanding 
masses  to  be  spheres.  Figs.  126  and  i2C.  show  the  results.  This 
is  not  merely  an  arbitrary  adjustment  of  curves  and  one  of  a  num- 
ber which  might  be  devised.  On  the  contrary,  it  has  been  shown 
in  the  discussion  of  Section  A  and  especially  in  Fig.  n  that  the  ratio 
of  the  two  maxima  of  the  deflection  components,  Fy  and  Fx,  and 
the  ratio  of  EEf  to  E'M  hold  a  definite  relation  to  the  distance  and 
depth  of  the  center  of  the  sphere.  Therefore  if  curve  2  be  drawn  in 
proper  proportion  and  as  shown  in  B  in  order  to  satisfy  the  demands 
of  the  y  component,  then  the  maximum  value  of  Fx  must  not  be 
over  40  per  cent  of  the  maximum  value  for  Fy,  even  if  the  center 


542  .  JOSEPH  BARRELL 

of  mass  is  close  to  the  surface.  It  may  be  any  value  less  than  40 
per  cent  of  maximum  Fy,  according  to  the  depth  of  the  center. 
But  having  chosen  that  ratio  which  appears  to  fit  the  demands  of 
the  data,  the  distance  of  the  point  of  maximum  Fx  from  the  point 
of  zero  value  becomes  also  fixed.  The  curves  numbered  2  in  Figs. 
i2B  and  C  must  therefore  satisfy  between  them  the  demands  of  the 
ratios  shown  in  Figs.  nE  and  F.  The  value  of  EE'  as  deduced 
from  either  curve  must  be  the  same. 

The  sum  of  all  the  Fy  curves  in  126  is  marked  S  Fy  and  must 
pass  through,  or  close  to,  the  points  which  measure  the  values 
given  by  the  deflection  residuals.  These  points  are  shown  as  small 
rectangles  in  B  and  C  and  give  ordinates  which  correspond  with 
the  size  of  the  components  of  the  residuals  as  shown  in  A.  In 
drawing  B  and  C  the  adjustment  of  the  curves  to  give  the  proper 
values  to  2  Fy  and  S  Fx  resulted  in  a  slight  readjustment  of  the 
centers  of  mass  as  shown  in  A.  The  positions  as  shown  in  Fig.  i2A 
have  been  determined  from  the  curves  below,  and  their  approximate 
agreement  with  the  initial  indications  of  the  resultants  is  a  check  on 
the  validity  of  the  solution.  It  is  seen  that  the  epicenter  of  a  mass 
should  not  lie  on  the  exact  intersection  of  any  two  resultants, 
since  at  the  point  of  measurement  several  masses  have  an  appreci- 
able influence  upon  the  direction  of  the  resultant.  The  adjustment 
of  the  curves  is  therefore  the  best  way  of  determining  finally  the 
best  location  of  the  epicenters. 

The  measurement  of  these  curves  gives  the  tabulation  of  data 
shown  in  Table  XXVIII. 

The  depth  to  the  centers  of  the  equivalent  spheres  having  been 
solved  by  means  of  the  ratios  given  in  Figs,  i  lE  and  i  iF,  the  masses 
of  these  spheres  are  ascertained  as  follows.  In  Fig.  n  the  value  of 
the  maximum  deflections  for  Fy  and  Fx  due  to  the  unit  sphere  are 
shown  for  various  distances  of  the  section  line  from  the  epicenter. 
For  example,  for  EE'=i.$D,  max.  Fy  is  4.6".  Now  for  sphere 
No.  4,  Fig.  i2A,  EE'=i.$D4  and  the  max.  Fy  is  6.2".  But  D 
for  the  unit  sphere  is  64  km.  whereas  D4  is  31  km.  Now  the  magni- 
tude of  the  deflections  for  points  similarly  situated  in  two  fields  of 
gravitative  force  will  vary  directly  as  the  respective  masses  and 
inversely  with  the  squares  of  the  distances.  This  may  be  put  into 


THE  STRENGTH  OF  THE  EARTH'S  CRUST 


543 


a  formula  as  follows:  Let  there  be  two  masses  M  and  Mn  with 
centers  at  depths  D  and  Dn ;  below  a  horizontal  plane.  For  points  on 
the  plane  similarly  situated  with  respect  to  their  respective  centers 
let  the  components  of  the  deflection  force  be  Fy  and  Fx  for  the  one, 
Fyn  and  Fxn  for  the  other.  Then 


Mn= 


_FynDn* 


FyD2 


M]     also  Mn  = 


FxnDn 
FxD2 


M 


The  results  of  the  application  of  this  formula  are  shown  in  the 
last  column  of  Table  XXVIII.  They  have  been  carried  out  to  two 
significant  figures,  but  the  second  is  not  to  be  regarded  as  accurate, 

TABLE  XXVIII 
INTERPRETATION  OF  RESIDUALS  IN  TERMS  OF  EQUIVALENT  SPHERES 


DATA 

RESULTS 

No.  OF 
SPHERE 

FROM  FIG.  12 

FIG.  nF 

FROM  FIG.  1  2 

FIG.  nE 

Max. 
Fy 

Max. 
Fx 

Max. 
Fy 

Max. 
Fx 

EE'  in 
Terms 
of  D 

EE' 

E'M 

EE' 
E'M 

EE'in 
Terms 
of  D 

DEPTH 
IN  KM. 

Mass  in  Terms  of 
Unit  Sphere 
Whose  R  =  $okm. 
d=o.i. 

i  

2  

S.i 
10.8 

2.0 
6.2 

8-4 
3-3 

2.3 
4-4 

2-5 

2.9 
3-5 
1-4 

.20 

.46 
.80 
•13 
.40 

•  35 

PPPPPP 

M  CO  O  W  «  <N 

42 
125 
13 

& 

34 

35 
95 
33 
4° 
67 
27 

1.20 
I.3I 
O.40 
I.I7 
I.3I 
1.27 

i.7D, 
2.5D* 
0.31)3 
i-5D< 
2.5DS 

2.2D6 

25 
40  to  50 
43 
31 

11 

+0.20M 

+2.5  to  2.gM 
+o.i7M 

—  0.32.J/ 

+i.ioM 
+o.o7M 

3  

5  
6  

even  if  the  original  data  are  accurate  to  the  second  place.  This  is 
because  the  error  of  the  square  of  a  quantity  is  approximately  twice 
as  great  as  the  original  error,  and  for  values  EE'  above  i .  oD  the 
error  in  even  the  first  power  of  D  is  appreciably  greater  than  the 
error  in  the  measured  quantities.  This  of  course  is  a  consideration 
of  the  error  in  the  determination  of  the  depth  and  mass  of  the 
hypothetical  spheres,  not  a  consideration  of  the  errors  in  the  deflec- 
tions themselves,  nor  related  to  the  fact  that  the  masses  are  in 
reality  not  spheres. 

It  has  been  shown  previously  that  the  interpretation  of  the 
outstanding  masses  in  terms  of  spheres  will  give  depths  too  great 
unless  the  real  masses  have  their  greatest  dimension  vertical.  For 
the  same  reasons  the  hypothetical  spheres  will  be  of  greater  mass 


544 


JOSEPH  BARRELL 


than  the  real  horizontally  extended  bodies  in  order  to  produce  the 
same  deflections  as  those  observed. 

The  question  arises  then  as  to  the  real  form  and  mass  of  the 
bodies  interpreted  here  as  spheres.  The  supplemental  data  at  hand 
yield  evidence  only  in  regard  to  No.  2.  This,  however,  is  the  great- 
est of  the  six  outstanding  masses  in  this  series.  The  supplemental 
data  consist  of  observations  on  gravity  anomalies  at  three  localities 
whose  distances  from  the  epicenter  of  No.  2  are  as  follows: 

RELATIONS  OF  GRAVITY  ANOMALIES  TO  THE  EPICENTER  OF  MASS  No.  2 

a,  47  km.  S.  E.          +0.031 

b,  163  km.  E.  —0.009 

c,  250  km.  N.N.W.     —0.029 

Now  the  curves  for  Fv  and  also  for  the  Fy  component  of  Fh  show 
with  increasing  distance  from  the  epicenter  a  rapid  fall  from  the 
maximum  value.  These  effects  of  the  gravitative  force  due  to  out- 
standing masses  are  consequently  markedly  local  and  the  nature 
of  the  mass  No.  2  must  have  a  distribution  such  as  to  account  for 
the  anomaly  of  +0.031  at  47  km.  from  the  epicenter.  The  inter- 
pretation must  not  give  a  form  which  will  exert  marked  influence 
upon  those  points  distant  163  and  250  km.  The  dominating  influ- 
ence of  mass  No.  2  on  the  value  of  Fv  appears  therefore  to  be  con- 
fined to  distances  within  125  or  150  km.  of  the  epicenter.  But  the 
two  limiting  spheres  of  mass  2 .  $M  and  2 .  gM  respectively  give  the 
relations  shown  in  the  first  two  lines  of  Table  XXIX. 

TABLE  XXIX 

INTERPRETATION  OF  MASS  No.  2 


Form 

Mass  in 
Terms  of 
Unit 
Sphere 

Depth  to 
Center  in 
Km. 

Radius  in 
Km. 

Height  in 
Km. 

Excess 
Density 
Above 
Mean 

Fv  at  Epi- 
center in 
Dynes 

Fv  at  47 
Km.  in 
Dynes 

A     Sphere 

2    C 

4O 

4O 

80 

O.  5 

o  $4.2 

0.  148 

B      Sphere 

2   0 

CQ 

c;o 

TOO 

o.  3 

O.4CX 

0.  1^7 

C      Cylinder 

2    1 

4O 

2OO 

IO   4 

O    I 

o  01  z 

D     Cylinder 

o  6 

2O 

TOO 

52 

O    2 

o  03  c. 

The  interpretation  of  the  deflections  as  due  to  spheres  gives  a 
gravity  anomaly  at  the  epicenter  of  the  sphere  between  four  and 
six  times  larger  than  the  largest  yet  observed  in  the  United  States. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  545 

At  the  distance  of  47km.  the  anomaly  would  be  from  0.148  to 
o.  157  dyne,  five  times  the  observed  value  of  0.031.  Clearly  then 
the  initial  interpretation  as  a  sphere,  although  it  satisfies  the  deflec- 
tion residuals  of  the  line  of  traverse,  is  far  from  the  truth.  The 
mass  must  have  horizontal  dimensions  much  greater  than  the 
vertical.  Assume  for  trial  that  the  mass  has  the  form  of  a  vertical 
cylinder  with  the  same  mass  and  depth  to  center  as  the  sphere, 
but  of  a  proportion  of  height  to  breadth  which  shall  satisfy  approxi- 
mately the  gravity  anomaly.  The  result  is  shown  in  the  third  line 
of  Table  XXIX.  The  gravity  anomaly  of  0.035  at  the  epicenter 
would  correspond  in  a  cylinder  of  these  proportions  to  a  value  only 
slightly  less  at  47  km.  But  the  radius  of  the  cylinder,  200  km.,  is 
now  far  too  great.  Other  cylinders  of  similar  form  will,  however, 
give  the  same  anomaly  at  the  epicenter  if  the  depth  and  dimensions 
are  all  divided  by  any  number,  n,  and  the  density  multiplied  by  the 
same  number.  This  gives  a  series  of  similar  cylinders  in  which  the 
density  varies  inversely  with  the  dimensions.  Of  such  a  series 
that  shown  in  the  fourth  line,  obtained  by  giving  n  a  value  of  2, 
comes  fairly  close  to  satisfying  all  the  requirements.  The  exact 
degree  of  adjustment  which  would  be  needed  to  satisfy  both  the 
gravity  anomaly  at  47  km.  and  the  deflections  on  the  line  of  traverse 
has  not  been  calculated.  If  this  were  done  and  the  dimensions 
adjusted  accordingly,  it  would  complete  a  second  approximation  to 
the  real  form  and  mass  of  No.  2.  Such  an  extended  treatment  of 
the  subject  would,  however,  be  beyond  the  immediate  purposes 
of  this  article  and  beyond  the  limits  of  space  which  it  should  occupy. 
The  data  also  are  at  present  hardly  of  a  sort  which  would  justify 
further  computations.  It  should  be  emphasized,  however,  that 
such  a  complete  investigation  is  not  difficult  and  would  require 
but  little  further  data,  properly  chosen,  to  check  the  conclusions. 

In  the  first  approximation,  the  mass  was  assumed  to  have  a  uni- 
form distribution  about  a  center,  giving  a  sphere.  In  the  second 
approximation,  the  vertical  axis  is  assumed  to  be  different  from  the 
horizontal  axes,  but  the  latter  being  kept  alike,  the  horizontal 
section  would  still  be  a  circle.  A  single  observation  of  the  gravity 
anomaly  near  the  epicenter  suffices  to  give  this  second  approxima- 
tion. The  third  approximation  would  be  to  consider  the  three 


546  JOSEPH  BARRELL 

co-ordinate  axes  of  the  mass  unlike.  This  would  require  some 
observations  in  two  directions  at  right  angles  from  the  epicenter. 
More  complete  data  consisting  of  both  deflection  and  gravity 
observations  would  of  course  give  still  closer  approximations  toward 
the  real  form  and  depth  of  the  mass.  What  it  is  desired  to  show 
here,  however,  is  that  the  interpretation  of  this  large  mass  as  a 
sphere  gave  a  depth  to  the  center  of  mass  about  twice  too  great  and 
a  mass  perhaps  four  times  too  great.  This  result  is  in  line  with  the 
general  deduction  previously  made  in  regard  to  the  direction  of 
the  error  involved  in  the  interpretation  of  deflection  residuals  as 
due  to  spheres.  It  contributes  its  individual  testimony  to  show 
that  the  masses  producing  the  notable  gravity  anomalies  and  deflec- 
tion residuals  are  situated  within  the  zone  of  isostatic  compensation 
and  more  especially  in  the  upper  part  of  that  zone. 

In  regard  to  the  large  positive  mass  No.  5  the  data  are  less 
determinative.  At  a  distance  of  60  km.  southeast  from  the  epi- 
center the  anomaly  is  only  +0.005  dyne.  At  150  km.  northeast  it 
is  —0.027  dyne.  It  thus  appears  that  there  are  some  large  nega- 
tive masses  easterly  of  No.  5.  As  this,  however,  is  on  the  side 
away  from  the  line  of  traverse  the  problem  of  the  real  form  and  mass 
of  No.  5  is  at  present  indeterminate. 

The  adjustment  between  the  deflection  curves  due  to  spherical 
masses  and  the  values  of  the  deflection  residuals  has  been  made 
closer,  perhaps,  than  the  probable  values  of  the  residuals.  Further- 
more, the  residuals  of  Solution  G,  if  they  had  been  given  for  this 
region,  would  have  required  a  somewhat  different  distribution  of 
masses.  A  solution  for  that  depth  of  compensation  which  would 
reduce  to  the  smallest  quantity  the  sum  of  the  least  squares  of  the 
residuals  of  this  area  1,000  km.  long  and  400  km.  wide  would  be 
still  somewhat  different.  That  local  solution  which  would  give  the 
smallest  residuals  would  be  such  as  would  make  small  the  algebraic 
sum  of  the  positive  and  negative  masses,  but  the  difference  in  mass 
between  positive  and  negative  centers  would  not  be  much  reduced. 
The  depth  to  the  centers  of  mass  would  be  the  quantity  most 
affected  by  a  change  in  the  hypothesis  regarding  the  depth  of  com- 
pensation. These  epicenters  then,  in  so  far  as  the  accidental  errors 
do  not  vitiate  the  values  of  the  residuals,  are  realities  in  nature. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  547 

In  view  of  this  analysis  of  the  data  given  in  Fig.  1 2  and  in  Table 
XXVIII,  it  is  to  be  concluded  that  for  this  region  even  the  larger 
outstanding  masses  from  Solution  H,  capable  of  exerting  a  notable 
influence  on  the  Fx  component  of  the  deflections  to  a  distance 
of  400  km.  or  more,  appear  to  have  their  centers  not  deeper  than 
20-25  km.  Their  mass  is  consequently  within  the  outer  half  or 
even  the  other  third  of  the  zone  of  isostatic  compensation  as  given 
by  Solution  H.  There  is  further  no  evidence  of  centrospheric 
heterogeneity. 

OTHER   INDICATIONS   REGARDING   DEPTH  OF   OUTSTANDING   MASSES 

Deflections  by  linear  or  dike-like  masses. — The  resultants  of  the 
deflection  residuals  are  shown  by  broken  lines  in  Fig.  12.  They 
show  a  tendency  to  converge  toward  centers.  This  is  true  in  general 
for  the  whole  United  States,  as  shown  in  Hayford's  illustration. 
This  tendency  to  convergence  indicates  that  the  dominating  out- 
standing masses  may  usually  be  regarded  in  a  first  or  second  approxi- 
mation as  symmetrical  with  respect  to  a  vertical  axis.  In  contrast, 
however,  to  this  rule  the  residuals  of  Solution  H  in  the  vicinity  of 
Washington1  indicate  an  outstanding  mass  with  a  northeast-south- 
west extension  of  at  least  120  km.,  whereas  the  breadth  is  probably 
not  more  than  20  km.  This  narrowness  is  shown  by  the  limited 
distance  between  the  large  residuals  of  opposite  sign  in  a  northwest- 
southeast  direction.  The  linear  extension  is  shown  by  the  parallel 
rather  than  radial  arrangement  of  the  resultants.  The  mass  gives 
rise  to  large  deflections  for  a  distance  of  as  great  as  100  km.  from 
the  sides,  but  its  influence  dies  out  somewhat  beyond. 

It  is  clear  from  these  relations  that  the  assumption  of  a  form  of 
the  mass  symmetrical  about  a  vertical  axis  for  the  purpose  of  deter- 
mining the  depth  of  the  center  would  not  be  justifiable.  Other 
assumptions  which  might  be  made  in  order  to  subject  the  mass  to 
mathematical  investigation  would  be  to  consider  it  as  a  horizontal 
prolate  spheroid,  or  as  a  horizontal  linear  mass  at  a  certain  depth, 
cylindriform,  or  as  a  vertical  plate.  The  latter  would  be  preferable. 
For  a  quantitative  solution  of  its  dimensions  and  mass  it  would 
be  desirable  to  have  some  observations  farther  to  the  northwest. 

1  Hayford,  Supplementary  Paper,  illustration  No.  4. 


548  JOSEPH  BAKRELL 

The  following  qualitative  conclusions  may,  however,  be  drawn  from 
an  inspection  of  the  residuals. 

Maximum  deflections  are  found  on  each  side  of  the  axial  line 
not  more  than  40  km.  distant.  The  deflections  continue  large  for 
at  least  twice  this  distance  but  not  for  three  times  this  distance. 
The  existence  of  large  residuals  so  close  to  the  axial  line  shows  con- 
clusively that  the  outstanding  mass  is  within  the  zone  of  compensa- 
tion and  apparently  within  its  outer  half,  but  the  maintenance  of  the 
size  of  the  deflections  without  much  change  for  a  considerably 
greater  distance  shows  also  that  it  is  not  merely  a  surficial  and 
linear  mass.  It  must  have  considerable  extension  in  depth.  In 
these  indications  it  agrees  therefore  with  the  more  precise  solution 
of  limiting  depth  given  for  the  Texas-Kansas  region. 

Indeterminate  evidence  from  anomaly  contours. — The  map  of 
anomaly  contours  shown  in  Fig.  5,  Part  II,  and  reduced  from  Bowie, 
does  not  in  general  throw  positive  light  on  the  depth  of  the  masses 
which  produce  the  anomalies  of  gravity.  The  necessarily  general- 
ized and  smoothed-out  character  of  this  map  has  been  discussed 
previously,  especially  in  Part  IV.  A  map  based  upon  more  numer- 
ous observations  would  show  higher  values  of  maximum  anomaly 
and  more  of  them.  The  centers  of  outstanding  mass  and  the 
anomaly  gradients  would  become  better  denned,  and  the  distances 
from  epicenter  to  half  value  of  Fv  would  average  smaller  than 
shown  at  present.  However,  notwithstanding  the  defects,  thirty- 
two  measurements  were  made  on  this  map  of  the  distances  from 
fifteen  pronounced  maxima  to  the  anomaly  contour  of  half  value, 
and  in  directions  not  toward  other  adjacent  maxima.  This  dis- 
tance was  chiefly  controlled  therefore  by  the  single  dominating 
mass.  The  measurements  gave  an  average  distance  of  120  km. 

If  the  outstanding  masses  which  gave  these  anomalies  were 
assumed  to  have  the  form  of  spheres,  this  would  give  their  centers 
a  depth  of  160  km.  and  imply  the  existence  of  marked  heterogeneity 
extending  below  the  zone  of  compensation  as  given  by  Solution  H. 
If  the  average  form  were  assumed,  however,  to  be  that  oblate  mass 
shown  in  Fig.  gC,  this  distance  to  the  contour  of  half  value  would 
correspond  to  a  depth  of  100  km.  But  such  assumptions  as  to 
form  are  hypothetical  and  justifiable  only  as  a  step  in  successive 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  549 

approximations,  not  as  a  conclusion.  The  anomalies  can  be 
accounted  for  just  as  readily  by  an  assumption  of  much  shallower 
depths.  The  outstanding  masses  would  then  possess  marked  thin- 
ness in  comparison  with  their  breadth.  They  would  be  hori- 
zontally extended  masses  or  the  algebraic  sum  of  many  masses;  in 
either  case  they  could  lie  within  a  quarter  of  the  depth  indicated  by 
the  assumption  of  spherical  form.  The  mere  measurement  of  the 
mean  distance  from  epicenter  to  half  value  of  Fv  is  therefore  wholly 
indeterminate  except  as  regards  the  limits  of  regional  compensation. 
In  some  respects,  however,  the  present  map  does  give  suggestions. 
Let  the  attention  be  turned  to  these  individual  features. 

A  line  of  stations  extends  along  the  margin  of  the  Coastal  Plain 
from  Washington,  D.C.,  to  Hoboken,  New  Jersey.  The  anomalies 
at  the  stations  and  their  distances  apart  as  measured  on  the  map 
are  as  follows: 

No.  22.    Washington,  D.C.,  +0.039  dyne 
58  km. 

No.  23.    Baltimore,  Md.,  — o.on 
138  km. 

No.  24.     Philadelphia,  Pa.,  +0.022 
6 1  km. 

No.  25.    Princeton,  N.J.,  —0.019 
69  km. 

No.  26.    Hoboken,  N.J.,  +0.024 

This  line  of  stations  extends  in  the  direction  of  the  trend  of  the 
foundation  rocks,  yet  the  sign  is  reversed  at  every  station,  showing 
marked  heterogeneity  even  in  the  direction  of  the  strike.  The 
average  anomaly  without  regard  to  sign  is  0.023,  a  little  larger  than 
the  average  for  the  whole  United  States.  The  average  with 
respect  to  sign  is  +-O.QII.  As  there  is  only  one  station  for  each 
of  the  positive  and  negative  masses  the  positions  and  magnitudes 
of  the  real  maxima  and  the  curves  of  changing  anomaly  are  unknown. 
Masses  in  the  upper  half  of  the  zone  of  compensation  could  produce 
these  effects  at  these  horizontal  distances  with  but  little  mutual 
neutralization.  Masses  below  the  zone  of  compensation  would, 
however,  have  to  be  very  great,  not  only  because  the  force  decreases 
inversely  with  the  square  of  the  distance,  but  because  masses  of  oppo- 
site sign  whose  centers  are  more  than  120  km.  deep  and  situated  but 


550  JOSEPH  BARRELL 

from  60  to  70  km.  apart  would  largely  neutralize  each  other  in 
their  surface  effects.  Furthermore,  there  is  no  notable  extension 
of  the  anomaly  contours  shown  in  any  direction,  and  more  espe- 
cially at  right  angles  to  the  line  of  stations,  such  as  would  suggest 
the  wider  fields  of  force  due  to  deep-seated  masses.  If  the  masses 
were  at  great  depth  this  limitation  of  attraction  to  regions  near  the 
epicenters  could  be  produced  only  by  a  special  checkerboard 
arrangement  of  opposite  masses  in  all  directions.  It  may  be 
rather  firmly  concluded,  therefore  that  the  anomalies  of  this  chain 
of  stations  along  a  line  of  low  topographic  relief  are  due  to  hetero- 
geneities of  density  within  the  zone  of  compensation. 

In  certain  regions,  as  in  Florida,  in  western  New  York  and 
Pennsylvania,  and  in  the  Great  Basin,  occur  broad  areas  of  anomaly 
showing  no  central  maximum.  To  some  extent  this  is  doubtless 
due  to  incompleteness  of  observations,  but  in  the  areas  mentioned 
the  stations  are  so  spaced  as  to  show  that  even  if  the  map  were 
complete  there  would  not  exist  marked  domes  of  anomaly,  such  as 
those  central  at  Minneapolis,  Minnesota,  and  at  Lead,  South 
Dakota.  This  absence  of  domal  form  of  anomaly  curves  suggests 
that  the  disturbing  masses  cannot  be  below  the  zone  of  compensa- 
tion, but  should  be  interpreted  as  due  to  the  effects  of  masses 
widely  distributed  in  the  zone  of  compensation.  This  relation  is 
especially  striking  in  southern  Nevada.  The  deflection  residuals  in 
northern  Utah  and  Nevada  all  turn  away  from  this  southern  area  of 
defective  mass,  shown  in  Fig.  5,  Part  II,  of  this  article,  as  located 
by  Hayford  and  Bowie.  Yet  within  this  broad  area  of  defective 
mass  Station  No.  67  shows  an  anomaly  of  only  —0.013  and  some 
of  the  surrounding  anomalies  have  actually  a  larger  negative  value. 
There  is  here  then  an  entire  absence  of  a  broad  domal  form.  This 
is  the  region  which  indicates  from  the  least-square  equations  of  the 
deflections  of  the  vertical  the  shallowest  compensation  within  the 
United  States;  and  the  combination  of  the  evidence  from  deflection 
residuals  and  anomaly  contours  goes  to  show  that  the  anomalies 
are  due  to  departures  from  isostasy  within  that  shallow  zone. 

An  inspection  of  Fig.  5,  Part  II,  shows  furthermore  that  the 
centers  of  plus  and  minus  attraction  as  located  by  Hayford  and 
Bowie  from  the  deflections  of  the  vertical,  although  in  general 


THE  STRENGTH  OF  THE  EARTH'S  CRUST       551 

agreement  with  the  measurements  of  gravity  anomalies,  so  far 
as  the  positive  or  negative  sign  of  the  center  of  mass  is  concerned, 
yet  are  not  closely  related  to  the  large  maxima.  They  are,  in  fact, 
in  most  cases  decidedly  eccentric  to  the  anomaly  contours.  The 
scarcity  of  these  areas  is  a  result  of  incompleteness  of  observations, 
but  their  eccentric  position  and  association  with  areas  of  moderate 
anomaly  is  not  an  error  due  to  the  reconnaissance  nature  of  the 
studies.  These  relations  indicate  that  the  neighboring  regions 
giving  broad  domal  areas  of  anomaly  are  in  such  cases  not  due  to  the 
dominating  control  of  centrospheric  heterogeneity,  for  in  that  case 
the  resultants  of  the  deflection  residuals  would  point  over  broad 
areas  in  the  general  direction  of  the  epicenter  of  the  mass.  The 
degree  of  discordance  between  the  centers  of  dominant  anomaly 
and  centers  of  dominant  deflection  indicates  that  fuller  observations, 
would  produce  agreement  by  adding  to  the  number  of  such  centers. 
The  present  data  suggest  therefore  that  the  areas  of  broad  excess 
or  defect  of  mass  as  shown  by  the  anomaly  map  are  due  to  aggre- 
gates more  or  less  composite  and  shallow,  so  that  each  part  influ- 
ences individually  to  some  extent  the  direction  of  the  deflection 
residuals  about  it.  Special  combinations  of  masses  of  shallow  depth 
with  other  masses  below  the  zone  of  compensation  could,  however, 
also  account  for  the  effects.  The  data  of  the  present  map  of 
gravity  anomalies  are  therefore  largely  indeterminate,  but  the  prob- 
abilities point  toward  at  least  the  greater  part  of  the  outstanding 
masses  lying  well  within  the  zone  of  compensation.  In  this  con- 
clusion the  data  agree  with  the  other  lines  of  evidence. 

RELATION   OF   DEPTH   OF    OUTSTANDING   MASSES    TO   HYPOTHESES 
REGARDING   DISTRIBUTION    OF   COMPENSATION 

The  measurements  of  the  deflection  residuals  are  very  much 
more  detailed  than  are  those  of  gravity  anomalies.  The  evidence 
from  them  is  rather  conclusive  that,  for  the  regions  investigated, 
the  excesses  or  defects  of  mass  which  cause  those  residuals  are 
situated  within  the  zone  of  compensation  and  more  especially  in  its 
outer  half  or  third.  Even  if  centers  of  outstanding  mass  were  uni- 
formly distributed,  however,  with  respect  to  depth,  they  would 
lose  influence  in  proportion  to  the  square  of  their  depth.  Smaller 


552  JOSEPH  BARRELL 

masses  which  would  exert  a  very  appreciable  effect  if  near  the 
surface  would,  in  consequence,  not  betray  their  existence  if  situ- 
ated near  the  base  of  the  zone  of  compensation.  But  the  larger 
masses  which  are  found  to  exist  would  exert  a  very  visible  control 
upon  the  deflections  of  the  vertical,  even  if  their  centers  were  at 
a  depth  of  100-200  km.  The  fact  that  such  depths  have  not  been 
found  suggests  that  the  larger  variations  from  the  mean  density 
within  any  one  earth  shell  tend  to  occur  in  the  outer  half  of  the 
zone  of  compensation  rather  than  in  its  deeper  parts  or  immedi- 
ately below  it. 

As  a  step  toward  the  interpretation  of  the  evidence,  let  the  con- 
clusion reached  in  Part  II  of  this  article  be  accepted :  that  regional 
isostasy  for  ordinary  relief  certainly  extends  to  a  radius  of  100  and 
probably  to  150  or  200  km.  Even  these  limits  do  not  reach  the 
capacity  of  crustal  strength.  Such  regional  limits  would  not  in 
reality  be  subject  to  sharp  boundaries.  This  agrees  with  the 
evidence  of  geology  in  showing  that  mountain  groups  of  circum- 
denudation — those  whose  relief  is  due  to  erosion  and  not  to  origi- 
nal differential  vertical  movement — are  upheld  by  the  rigidity  of 
the  crust.  This  applies  to  many  of  the  mountain  groups  of  the 
Appalachians;  such,  for  example,  as  the  Catskills. 

The  fairest  initial  hypothesis  of  isostatic  compensation  would  be 
then  to  calculate  for  each  station  the  average  elevation  of  the 
country  within  a  radius  of  99  km.,  being  the  outer  radius  of  zone  N, 
and  to  assume  a  uniform  density  to  these  limits  such  as  is  needed  to 
compensate  this  area.  A  second  trial  hypothesis  would  be  to  use 
j,s  the  radius  of  regional  compensation  the  outer  limits  of  zone  0, 
166.7  km.  Under  these  two  calculations  for  regional  compensa- 
tion the  Catskills  would  be  regarded  as  producing  deflections 
which  should  show  an  excess  of  mass  at  the  surface  of  the  earth. 
Such  an  erosion  basin  as  the  Nashville  basin  should  show,  on  the 
other  hand,  by  its  deflections  a  surface  deficiency  of  mass.  For 
the  hypothesis  which  approaches  nearest  to  the  truth,  the  residuals 
of  the  deflections  should  be  small  and  the  outstanding  masses 
would  be  determined  by  variations  of  density  within  the  crust  and 
not  of  the  topography  upon  its  surface. 

Under  the  hypothesis  of  local  compensation  as  given  in  Solution 
H  the  excess  of  mass  in  the  Catskills  would  show,  on  the  contrary, 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  553 

as  a  slightly  excessive  density  throughout  the  whole  zone  of  com- 
pensation; the  Nashville  basin  as  a  slightly  deficient  density 
through  the  same  depth.  The  residuals  should  indicate  an  out- 
standing excess  and  deficiency  of  mass  respectively  with  the  centers 
at  a  depth  near  the  middle  of  the  zone  of  compensation.  But 
masses  with  centers  at  this  depth  and  distance  would  have  a  very 
diminished  maximum  effect  upon  the  residuals  of  the  deflections 
of  the  vertical,  and  one  largely  modified  by  the  effects  of  con- 
tiguous regions.  Heterogeneities  of  density  nearer  the  surface 
and  not  related  to  compensation  would  tend  also  to  overshadow  the 
error  involved  in  the  hypothesis  of  local  compensation.  It  would 
appear  then  that  the  nature  of  the  deflections  is  not  very  sensitive 
for  testing  the  relative  probability  of  the  hypotheses  of  local  yersus 
regional  compensation.  The  assistance  of  a  computing  office 
for  trying  out  several  hypotheses  would  probably  bring  to  light, 
however,  conclusions  which  would  be  more  determinative.  These 
statements  must  be  regarded,  therefore,  as  forecasts  not  yet  sub- 
jected to  the  tests  of  computation. 

In  view  of  the  preceding  discussion  it  would  seem  that  the 
deflection  residuals  of  Solution  H  are  chiefly  of  value  for  measuring 
the  heterogeneities  of  density  not  related  to  topography,  nor  to 
the  mantle  of  sedimentary  rocks.  This  is  especially  true  of  the 
Texas-Kansas  region  studied  in  detail,  for  there  the  region  is  one 
of  plains  with  an  average  elevation  of  about  a  thousand  feet,  and 
the  demands  of  local  isostasy  as  postulated  in  Solution  H  would 
call  for  a  nearly  uniform  density  under  all  this  region.  The  out- 
standing masses  represent  in  large  part,  therefore,  real  and  local 
variations  from  a  mean  density  of  the  continental  crust. 

But  if  masses  of  excess  or  defect  of  density  similar  to  those  num- 
bered 2  and  5  of  Fig.  12  were  widely  extended,  say  to  a  radius  of 
500  rather  than  100  km.,  they  would  tend  much  more  strongly  to 
make  for  a  local  or  intracontinental  isostatic  adjustment.  They 
would  become  then  not  outstanding  masses  but  in  large  part  com- 
pensating masses.  The  outstanding  masses  represent  the  same 
kind  of  variations,  therefore,  which  if  more  broadly  extended  would 
be  in  accord  with  an  isostatic  adjustment  of  topography  to  a  differ- 
ent level.  They  suggest  that  if  the  zone  of  compensation  of  the 
continental  crust  be  divided  into  three  shells  of  40  km.  each  in 


554  JOSEPH  BARRELL 

thickness,  the  greatest  variations  in  density  take  place  in  the  outer 
shell.  This  conclusion  should  be  regarded  as  tentative,  however, 
until  confirmed  by  wider  detailed  studies  and  more  numerous 
examples. 

Accepting  for  the  present  this  tentative  conclusion,  how  does 
it  agree  with  that  previously  reached — that  isostatic  compensation 
in  some  regions  appears  to  go  notably  deeper  than  122  km.  and  that, 
where  deep,  the  residuals  average  smaller  than  for  the  continent  in 
general  ?  The  answer  would  appear  to  be  that  moderate  variations 
of  density  are  sufficient  to  account  for  the  isostatic  relations  of 
different  parts  of  the  continent  to  each  other  and  that  these  moder- 
ate variations  may  go  very  deep. 

If  the  actual  distribution  of  compensation  gradually  disappears 
with  depth,  the  hypothesis  of  uniform  compensation  complete  at  a 
certain  depth  corresponds  to  two  outstanding  masses,  one  just  above 
the  limiting  surface,  122  km.  in  Solution  H,  the  other  just  below 
that  surface.  But  these  masses  would  largely  balance  each  other, 
having  opposite  signs;  so  that  they  would  give  at  the  surface  of  the 
earth  but  little  evidence  of  their  existence.  Imperfections  of  the 
hypothesis  in  regard  to  the  bottom  of  the  zone  of  compensation 
would  in  consequence  not  readily  be  detected  by  methods  for 
determining  the  depth  of  outstanding  masses. 

The  isostatic  balance  of  continental  crust  against  oceanic  crust 
is  a  somewhat  different  problem  from  that  of  the  different  segments 
of  the  continent  with  respect  to  each  other.  Solution  H  requires 
a  mean  difference  in  specific  gravity  of  about  o.i  to  a  depth  of 
122  km.  between  the  crust  of  the  average  continental  and  average 
oceanic  segments.  The  contrasts  in  density  are  therefore  pro- 
nounced and  go  very  deep.  Within  the  continent,  on  the  other 
hand,  the  variations  in  density  related  to  isostatic  compensation 
are  comparatively  small  and  this  investigation  suggests  that  those 
variations  may  be  more  largely  in  the  higher  levels  of  the  crust. 

In  conclusion,  the  depths  of  the  outstanding  masses  are  seen 
to  be  related  to  many  problems  in  crustal  statics  and  dynamics. 
The  depth  determines  the  magnitude  of  the  masses  involved  and  if 
known  will  serve  as  a  test  of  various  hypotheses.  The  excesses 
and  defects  of  mass  departing  from  that  mean  which  is  demanded 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  555 

by  the  best  hypothesis  will  be  a  more  accurate  measure  of  the 
capacity  of  the  rigid  crust  to  carry  without  viscous  yielding  loads 
which  have  been  borne  through  geologic  time,  hidden  loads  whose 
magnitudes  in  many  regions  appear  to  mask  by  contrast  the  present 
relief  between  mountains  and  valleys. 

The  measure  and  the  meaning  of  the  variable  distribution  of 
mass  within  the  lithosphere  constitutes  an  inviting  field  of  geology, 
discernible  in  the  present,  but  whose  real  exploration  is  a  work  of 
the  future. 

[To  be  continued] 


Reprinted  from  THE  JOURNAL  or  GEOLOGY,  Vol.  XXII,  No.  7,  Oct.-Nov.  1914 


THE  STRENGTH  OF  THE  EARTH'S  CRUST 


JOSEPH  BARRELL 
New  Haven,  Connecticut 


PART  VI.     RELATIONS   OF  ISOSTATIC   MOVEMENTS   TO 
A  SPHERE  OF  WEAKNESS— THE  ASTHENOSPHERE.' 

INTRODUCTION  AND  SUMMARY 655 

STRESS-DIFFERENCES  BETWEEN  CONTIGUOUS  COLUMNS  OF  THE  CRUST  659 

Stresses  under  Conditions  of  Isostatic  Equilibrium      ....  659 

Modifications  of  Stresses  Produced  by  Base-Leveling         .        .        .  666 

Relief  of  Stress  Accompanying  Restoration  of  Isostasy      .       .       .  670 

RELATIONS  OF  UNDERTOW  TO  THE  ZONE  OF  COMPENSATION  .       .       .  672 

Present  Status  of  the  Problem 672 

Objections  against  Undertow  in  the  Zone  of  Compensation       .        .  677 

Undertow  Restricted  to  a  Sphere  of  Weakness — the  Asthenosphere  680 

INTRODUCTION   AND   SUMMARY 

In  studies  on  the  nature  of  isostasy  it  is  necessary  to  distinguish 
between,  first,  the  existence  of  isostasy;  second,  the  limits  of 
isostatic  equilibrium;  and  third,  the  mode  of  maintenance  of  this 
equilibrium. 

The  first  has  long  been  known,  the  knowledge  of  the  existence 
of  some  relation  of  density  counterbalancing  elevation  having  been 
gradually  developed  since  the  middle  of  the  nineteenth  century 
through  the  determination  of  the  local  deviations  of  the  vertical 
as  shown  by  the  comparison  of  the  astronomic  and  geodetic  latitudes 
for  the  same  station.  This  was  a  problem  which  arose  in  both 
astronomy  and  geodesy.  It  was  found,  when  the  attractive  effect 
of  the  mountain  regions  was  computed,  that  they  did  not  deflect 
the  vertical  at  adjacent  stations  as  much  as  was  to  be  expected 
from  their  visible  masses.  The  phenomenon  was  first  pointed  out 

1  An  abstract  of  Parts  V,  VI,  and  VII  of  this  series  was  given  at  the  April,  1914, 
meeting  of  the  American  Philosophical  Society  at  Philadelphia  under  the  title, 
"Relations  of  Isostasy  to  a  Zone  of  Weakness — the  Asthenosphere."  See  Science, 
XXXIX,  842. 

655 


656  JOSEPH  BARRELL 

by  Petit  in  I849-1  Archdeacon  Pratt  of  Calcutta  showed  a  few 
years  later  that  whereas  a  discrepancy  of  5 .  2"  existed  between  the 
geodetic  and  astronomic  latitudes  of  Kalianpur  and  Kaliana,  the 
calculation  of  the  effect  of  the  Himalayas  called  for  a  difference  of 

IS-  9"-* 

These  facts  were  definitely  formulated  into  a  theory  of  isostasy 
by  the  Astronomer  Royal  of  Great  Britain,  G.  B.  Airy,  within  a 
year  following  the  appearance  of  Pratt's  paper,3  though  it  remained 
for  Button  to  recognize  the  large  geologic  significance  and  to  coin 
for  the  relations  of  elevation  and  density  the  word  isostasy.4  Fol- 
lowing this  Putnam  and  Gilbert  showed  by  gravity  measurements 
that  a  considerable  degree  of  regional  isostasy  existed  over  the 
United  States.5  Since  then  has  appeared  the  much  more  detailed 
work  of  Hay  ford  and  Bowie,  the  computations  made  by  the  comput- 
ing office  of  the  United  States  Coast  and  Geodetic  Survey  under 
their  directions  making  possible  this  present  investigation. 

Thus  there  has  developed  through  more  than  half  a  century 
evidence  beyond  controversy  which  shows  that  the  earth's  crust 
in  its  larger  relief  and,  within  certain  limits,  even  its  smaller  features, 
such  as  the  great  plateaus  and  basins,  rests  more  or  less  approxi- 
mately in  flotational  equilibrium. 

The  second  division  of  the  larger  problem  of  isostasy,  that  of  the 
areal  limits  and*  degree  of  perfection  of  isostatic  adjustment,  is  the 
subject  which  has  been  dealt  with  in  the  previous  parts  of  this 
investigation.  It  has  been  found  that,  although  the  relations  of 
continents  and  ocean  basins  show  with  respect  to  each  other  a  high 

1  "Sur  la  latitude  de  1'Observatoire  de  Toulouse,  la  densite  moyenne  de  la  Chaine 
des  Pyrenees,  et  la  probabilite  qu'il  existe  uh  vide  sons  cette  chaine,"  Comptes  rendus 
de  I'Acad.  des  Sc.,  XXIX  (1849),  73Q. 

2  "On  the  Attraction  of  the  Himalaya  Mountains  and  of  the  Elevated  Regions 
beyond  Them,  upon  the  Plumbline  in  India,"  Phil.  Trans.  Roy.  Soc.,  Vol.  CXLV 
(i8SS). 

3  G.  B.  Airy,  "On  the  Computation  of  the  Effect  of  the  Attraction  of  Mountain 
Masses  as  Disturbing  the  Apparent  Astronomical  Latitude  of  Stations  in  Geodetic 
Surveys,"  Phil.  Trans.  Roy.  Soc.,  Vol.  CXLV  (1855). 

4  "On  Some  of  the  Greater  Problems  of  Physical  Geology,"  Bull.  Phil.  Soc.  Wash., 
XI  (1889),  53. 

3  Bull.  Phil.  Soc.  Wash.,  XIII  (1895),  31-75. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  657 

degree  of  isostasy,  there  is  but  little  such  adjustment  within  areas 
200  to  300  km.  in  diameter,  or  of  limited  differential  relief.  Indi- 
vidual mountains  and  mountain  ranges  may  stand  by  virtue  of  the 
rigidity  of  the  crust.  Even  under  the  level  plains  equally  great 
loads  are  permanently  borne,  loads  produced  by  widespread  irregu- 
larities of  density  not  in  accord  with  the  topography  above. 
Isostasy,  then,  is  nearly  perfect,  or  is  very  imperfect,  or  even  non- 
existent, according  to  the  size  and  relief  of  the  area  considered. 

The  third  division,  the  mode  of  maintenance  of  isostasy  and  its 
bearings  on  problems  of  the  crust,  remains  to  be  considered.  This 
condition  of  isostatic  equilibrium  exists  at  present  in  spite  of  the 
leveling  surface  actions  and  compressive  crustal  movements  of  all 
past  geologic  time.  There  must  be,  consequently,  some  internal 
mode  of  restoring  more  or  less  perfectly  an  isostatic  condition, 
either  by  frequent  small  movements,  or  by  more  infrequent  and 
larger  ones. 

Erosion  and  sedimentation  result  in  a  lateral  transfer  of  matter, 
and  to  maintain  isostasy  there  must  be  some  lateral  counter- 
movement  in  the  earth  below,  but  in  regard  to  how  or  where  or 
when  this  is  done,  and  as  to  what  are  its  effects,  there  has  been  no 
unanimity  of  opinion,  nor  convincing  demonstration. 

In  considering  the  problems  of  crustal  dynamics  some  authors 
have  regarded  earth  shrinkage  and  consequent  tangentially  com- 
pressive forces  as  controlling  the  nature  of  diastrophism,  including 
movements  of  both  orogenic  and  epeirogenic  character;  others,  the 
advocates  of  extreme  isostasy,  have  thought  to  see  even  in  folding 
only  the  secondary  effects  of  movements  maintaining  isostatic 
equilibrium.  The  first  point  of  view  emphasizes  the  strength  and 
elasticity  of  the  crust,  with  long-deferred  periodic  discharge  of 
stress.  The  second  point  of  view  calls  for  an  interpretation  based 
on  the  weakness  and  plasticity  of  the  crust,  with  resulting  nearly 
continuous  small  movements  restoring  the  delicate  vertical  balance 
destroyed  by  gradational  actions.  To  what  degree  are  the  two 
points  of  view  compatible  and  within  what  limits  is  each  dominant  ? 
The  problem  of  this  chapter  involves,  therefore,  not  only  the  mode 
but  the  limits  and  effects  of  the  movements  which  more  or  less 
completely  maintain  or  restore  isostasy. 


658  JOSEPH  BARRELL 

The  method  of  attack  is  largely  one  of  exclusion.  By  showing 
what  hypotheses  cannot  apply,  the  way  is  prepared  for  conclusions 
in  better  accord  with  the  fields  of  fact  and  theory. 

The  results  show  that  conditions  of  isostatic  equilibrium  cause 
the  light  and  high  segments  to  press  heavily  against  the  adjacent 
lower  and  heavier  ones,  most  heavily  above.  The  tendency  is 
consequently  for  the  high  areas  to  spread  with  a  glacier-like  flow 
over  the  low  areas.  This  tendency,  however,  is  effectively  resisted 
by  the  strength  of  the  crust.  Upon  the  disturbance  of  equilibrium 
by  erosion  and  deposition  there  are  two  kinds  of  stresses  produced 
which  tend  to  restore  equilibrium.  The  first  is  a  tendency  of  the 
heavy  column  to  underthrust  the  lighter,  but  it  could  never  produce 
compression  and  folding  at  the  surface.  This  force  would  be  most 
effective  under  the  hypothesis  of  great  crustal  weakness,  so  that 
the  vertical  stresses  could  be  transmitted  in  a  horizontal  direction 
within  the  lithosphere  as  in  a  fluid.  Even  in  that  case,  however, 
it  would  not  be  the  dominating  force.  The  actual  isostatic  move- 
ments consist  of  a  rising  of  the  eroded  areas,  a  sinking  of  those 
which  are  loaded.  This  involves  shear  or  flexure  around  their 
boundaries.  The  columns  must  be  large  enough  so  that  the  excess 
or  deficiency  of  mass  can  become  effective  in  producing  deformation. 
When  the  accumulating  vertical  stresses  have  overcome  the  strength 
of  the  crust,  the  excess  pressure  from  the  heavy  area  is  transmitted 
to  the  zone  below  the  level  of  compensation.  This  deep  zone  is  in 
turn  the  hydraulic  agent  which  converts  the  gravity  of  the  excess 
of  matter  in  the  heavy  column  into  a  force  acting  upward  against 
the  lighter  column  and  thus  deforms  the  crust  of  the  eroded  area. 
By  this  means  even  the  continental  interiors  are  kept  in  isostatic 
equilibrium  with  the  distant  ocean  basins.  This  implies  a  great 
depth  and  thickness  to  the  zone  of  plastic  flow.  Although  it  must 
be  plastic  under  moderate  permanent  stresses,  this  does  not  imply 
by  any  means  a  necessarily  fluid  condition,  and  fluidity  is  disproved 
by  other  lines  of  evidence. 

The  zone  of  compensation,  being  competent  to  sustain  the 
stresses  imposed  by  the  topography  and  its  isostatic  compensation, 
must  obey  the  laws  pertaining  to  the  elasticity  of  the  solid  state 
and  is  to  be  regarded  therefore  as  of  the  nature  of  rock.  Conse- 
quently there  may  be  extended  to  all  of  it  the  name  of  the  litho- 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  659 

sphere,  even  though  it  includes  from  time  to  time  molten  bodies, 
the  constituents  of  the  pyrosphere. 

The  theory  of  isostasy  shows  that  below  the  lithosphere  there 
exists  in  contradistinction  a  thick  earth-shell  marked  by  a  capacity 
to  yield  readily  to  long-enduring  strains  of  limited  magnitude. 
But  if  such  a  zone  exists  it  must  exercise  a  fundamental  control 
in  terrestrial  mechanics,  in  deformations  of  both  vertical  and  tan- 
gential nature.  It  is  a  real  zone  between  the  lithosphere  above 
and  the  centrosphere  below,  both  of  which  possess  the  strength  to 
bear,  without  yielding,  large  and  long-enduring  strains.  Its  reality 
is  not  lessened  because  it  blends  on  the  limits  into  these  neighbor- 
ing spheres,  nor  because  its  limits  will  vary  to  some  degree  with  the 
nature  of  the  stresses  brought  upon  it  and  to  a  large  degree  by 
the  awakening  and  ascent  of  regional  igneous  activity.  To  give 
proper  emphasis  and  avoid  the  repetition  of  descriptive  clauses  it 
needs  a  distinctive  name.  It  may  be  the  generating  zone  of  the 
pyrosphere;  it  may  be  a  sphere  of  unstable  state,  but  this  to  a 
larger  extent  is  hypothesis  and  the  reason  for  choosing  a  name 
rests  upon  the  definite  part  it  seems  to  play  in  crustal  dynamics. 
Its  comparative  weakness  is  in  that  connection  its  distinctive 
feature.  It  may  then  be  called  the  sphere  of  weakness — the 
asthenosphere,  and  its  position  among  the  successive  shells  which 
make  up  the  body  of  the  earth  is  as  follows: 

The  atmosphere 

Including  the  biosphere 
The  hydrosphere 

The  lithosphere 

Including  the  pyrosphere 
The  asthenosphere 

The  centrosphere,  or  barysphere 

Each  has  played  its  fundamental  part  in  the  development  of 
earth-history. 

STRESS-DIFFERENCES  BETWEEN  CONTIGUOUS  COLUMNS  OF  THE  CRUST 

Stresses  under  conditions  of  isostatic  equilibrium.— The  conti- 
nental platforms  slope  down  into  the  ocean  basins  at  grades  which 
range  mostly  from  one  in  ten  to  one  in  thirty.  Some  of  the  great 


660  JOSEPH  BARRELL 

foredeeps  show  both  the  greatest  depths  of  water  and  the  steepest 
descents.  The  Chilean  coast,  for  instance,  at  lat.  25°  S.,  slopes 
from  the  Andes  to  a  depth  of  7,500  meters  with  a  submarine  grade 
of  one  in  eight.  Under  the  hypothesis  of  nearly  perfect  isostasy, 
which  will  be  favored  in  this  discussion,  this  would  be  taken  to 
show  the  contiguity  of  areas  in  the  crust  of  markedly  unlike  density. 

Let  the  slope  between  such  areas  be  regarded  as  a  thick  parti- 
tion between  two  columns,  each  in  isostatic  equilibrium.  These 
rest  then  upon  the  substratum  below  the  zone  of  compensation 
with  the  same  pressure  and  stand  vertically  in  equilibrium. 

In  so  far  as  the  rock  within  the  crust  is  subjected  to  mere  cubic 
compression,  equal  in  all  directions  and  increasing  with  depth, 
there  is  no  distortional  force.  In  so  far,  however,  as  side  pressures 
in  one  column  are  not  balanced  by  equal  side  pressures  from  the 
adjacent  columns,  there  is  a  stress-difference  which  does  produce 
a  distortional  strain.  If  the  stress-difference  exceeds  the  elastic 
limit  a  permanent  deformation  results  which  reduces  the  stress 
and  eases  the  strain.  It  is  the  plan  of  this  paper  to  discuss  the 
nature  of  the  stress-differences  on  the  partition  separating  two 
contiguous  columns  of  the  crust,  of  markedly  unlike  density; 
first,  when  these  are  in  isostatic  equilibrium,  and  second,  when 
not  in  such  equilibrium.  Fig.  13  is  drawn  to  show  graphically 
these  relations. 

The  land-column  of  the  crust  is  marked  M;  the  submarine 
column  is  N;  0  is  the  earth-shell  below  the  zone  of  isostatic  com- 
pensation; P  is  the  column  of  sea- water.  The  vertical  partition 
between  the  unlike  columns  stops  in  reality,  according  to  the  hy- 
pothesis, at  the  bottom  of  the  columns.  It  is  here  extended  down 
through  the  earth-shell  O-O  in  order  to  discuss  the  deformation 
which  would  take  place  in  the  latter  shell.  M  and  N  represent 
what  is  here  called  the  lithosphere;  O-O  the  zone  which  it  is 
proposed  to  call  the  asthenosphere. 

In  case  A,  isostatic  equilibrium  is  assumed  and  the  pressures 
of  the  two  lithospheric  columns  are  equal  upon  the  asthenosphere. 
But,  assuming  for  the  moment  that  the  vertical  pressures  are  freely 
transmitted  as  lateral  pressures,  it  is  seen  that  a  marked  horizontal 
unbalanced  pressure  is  produced  by  the  land-column  against  the 


THE  STRENGTH  OF  THE  EARTH'S  CRUST 


661 


sea-column,  as  represented  by  the  horizontal  lines  of  the  stress 
diagram.  The  top  of  the  land-column  is  balanced  only  against  the 
negligible  weight  of  the  atmosphere  and  the  lateral  stress  gradient 
is  there  highest.  The  next  portion  below  is  balanced  against  the 


Sea  -level 


Stress  scale 
kg.persqcm. 
Pressures 
avoumedas 

Densities 

1?70:N2.T/ 


FIG.  13. — Diagram  illustrating  pressure-relations  of  the  crust  for  marginal 
portions  of  the  continental  shelf  and  oceanic  basin,  interpreted  as  balanced  by  uni- 
formly distributed  isostatic  compensation.  Stress-differences  are  shown  by  cross- 
lined  diagrams,  the  pressures  being  regarded  as  transmitted  hydrostatically.  The 
actual  lateral  stress-differences,  for  stresses  within  the  elastic  limit,  are  about  one- 
fourth  of  the  hydrostatic  pressures  here  shown. 

A.  Columns  in  isostatic  equilibrium. 

B.  Relations  after  base-leveling. 

C.  Relations  after  re-establishment  of  isostatic  equilibrium. 


662  JOSEPH  BARRELL 

sea-water  and  the  stress  gradient  becomes  less  high.  The  maximum 
thrust  occurs  at  the  bottom  of  the  ocean  and  is  from  the  land  toward 
the  sea.  Below  this  level  the  density  of  the  sea-column  is  greater 
than  that  of  the  land-column.  This,  with  increasing  depth, 
gradually  balances  the  excess  pressure,  and  at  the  base  of  the  litho- 
sphere  both  the  lateral  and  vertical  pressures  of  both  columns  by 
hypothesis  are  equal. 

In  this  diagram  the  pressures  of  the  columns  are  imagined  to 
act  hydrostatically,  but,  in  reality,  for  stresses  within  the  elastic 
limit,  this  would  not  be  so.  Further,  in  so  far  as  the  partition  is 
much  wider  than  the  difference  in  elevation  of  the  columns,  it  has 
a  gentle  surface  slope  and  will  tend  to  give  the  upper  part  of  the 
land-column  competence  to  hold  itself  in  by  its  own  strength  and 
that  of  the  partition.  The  approximate  ratio  which  the  actual 
lateral  pressure-differences  on  the  two  sides  of  the  partition  hold  to 
the  assumed  hydrostatic  pressures  may  be  perceived  from  the 
results  of  a  recent  work  by  Love  entitled  Some  Problems  of  Geo- 
dynamics*  In  chaps,  ii  and  iii  he  considers  the  problems  of  the 
isostatic  support  of  continents  and  mountains.  As  a  basis  for  the 
analytic  treatment  he  assumes,  first,  the  existence  of  complete  com- 
pensation within  a  depth  of  one-fiftieth  of  the  earth's  radius, 
127  km.;  second,  that  at  this  depth  all  stress-differences  disappear, 
the  pressures  below  being  of  the  nature  of  hydrostatic  pressures, 
the  only  kind  which  could  occur  if  a  fluid  layer  existed  at  and  below 
the  depth  of  127  km.;  third,  it  is  known  that  the  heterogeneities  of 
mass  in  the  lithosphere  only  slightly  modify  the  form  of  the  geoid, 
and  it  is  accordingly  assumed  that  there  is  no  such  effect.  Love 
thus  treats  of  the  limiting  case  of  a  crust  exhibiting  perfect  isos- 
tasy,  its  surface  relief  not  modifying  the  form  of  the  geoid  given  by 
the  ocean  surface,  and  resting  with  its  base  upon  a  fluid  zone.  As 
such,  his  solution  is  of  great  value,  but  he  states:  "It  must,  how- 
ever, be  understood  that  the  special  form  (of  the  hypothesis  of 
isostasy)  is  introduced  for  the  sake  of  analytical  simplicity  rather 
than  physical  appropriateness."2 

The  artificiality  of  the  assumption  of  the  existence  of  no  -stress- 
differences  below  the  zone  of  compensation  is  shown  by  the  law  of 

1  Cambridge  University  Press,  1911.  2  Op  cit.,  p.  7. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  663 

density  distribution  which  results.  With  only  these  three  limiting 
assumptions,  the  number  of  unknown  quantities  remains  larger 
than  the  number  of  equations,  and  the  results  are,  strictly  speaking, 
indeterminate;  but  by  making  various  reasonable  further  assump- 
tions definite  solutions  in  accordance  with  these  may  be  obtained. 
The  elimination  of  stress-differences  at  the  base  of  the  lithosphere, 
taken  as  equivalent  here  to  the  zone  of  compensation,  requires, 
however,  that  there  shall  be  a  peculiar  relation  of  densities.  To 
compensate  an  elevation  it  must  be  offset  by  matter  below  of  less 
density  than  the  mean  for  that  depth,  but  in  order  to  quench  the 
stress-differences  at  the  base  of  the  lithosphere  there  must  be 
between  the  light  matter  and  this  base  a  layer  of  more  than  mean 
density  for  that  depth.  Thus  the  light  layer  must  perform  a  two- 
fold function,  compensating  not  only  the  elevation  above  but  the 
heavy  layer  below.  For  depressions  in  the  crust  there  must  be  a 
reverse  arrangement,  matter  of  more  than  mean  density  existing 
immediately  below  the  surface.  But  above  the  base  of  the  litho- 
sphere there  must  be  a  layer  of  less  than  mean  density  for  that 
depth.  The  artificialities  of  this  scheme  would  be  sufficient  to 
form  a  disproof  of  the  initial  assumption  which  determined  it,  but 
it  also  seems  to  be  directly  disproved  by  the  evidence  brought 
forward  in  the  earlier  parts  of  the  present  article.  Nevertheless, 
the  exact  mathematical  solution  of  this  difficult  problem  is  of  great 
value  as  giving  the  results  of  the  assumptions  of  extreme  isostasy. 

For  the  largest  inequality  of  the  crust,  regarded  as  a  zonal 
harmonic  of  the  first  order,  that  represented  by  the  land  and  water 
hemispheres,  Love  shows  that  the  lateral  stress-differences  under 
this  hypothesis  of  isostasy  reach  a  maximum  at  a  depth  equal  to 
one- third  of  the  zone  of  compensation  and  are  equal  to  only  o .  006 
of  the  weight  of  a  column  of  rock  of  height  equal  to  the  maximum 
height  of  the  inequality.  For  harmonics  of  the  second  and  third 
orders,  representing  the  continents,  the  fractions  are  0.0134  and 
0.0208.  These  results,  Love  states,  are  extremely  favorable  to  the 
hypothesis  of  isostasy,  since  the  inequalities  could  be  supported  by 
any  reasonably  strong  material. 

There  are  two  criticisms,  however,  to  be  noted  while  citing  this 
conclusion.  First,  it  is  known  that  the  crust  is  vastly  stronger  than 


664  JOSEPH  BARRELL 

these  requirements,  so  that  such  a  perfected  isostatic  arrangement 
is  not  demanded  on  the  score  of  crustal  weakness.  Second,  the 
harmonic  curves  giving  these  figures  are  of  a  gently  sweeping 
character;  whereas,  the  actual  continents  are  in  many  places  high 
on  their  margins,  and  from  these  margins  they  slope  with  compara- 
tive steepness  to  the  mean  depth  of  the  ocean  floors.  The  stresses 
set  up  beneath  the  continental  margins  are  accordingly  a  closer 
approximation  to  those  imposed  by  lofty  mountain  ranges.  Assume 
that  compensations  of  the  continental  margins  are  perfect  and  the 
problem  becomes  that  which  Love  takes  up  in  the  following  chap- 
ter, namely,  the  isostatic  support  of  mountains,  except  that  we  deal 
with  only  one  great  slope,  whereas  the  theory  calls  for  a  succession 
of  mountains  and  valleys. 

It  is  shown  that  for  such  a  compensated  series,  postulating  the 
distribution  of  densities  previously  discussed,  the  greatest  stress- 
difference  exists  at  the  mean  surface,  beneath  the  crests,  and 
reaches  a  value  equal  to  half  the  weight  of  a  column  of  rock  equal 
to  half  the  height  of  the  crests  above  the  valley  bottoms.  From 
this  maximum  the  stress-difference  decreases  to  zero  at  the  base  of 
the  zone  of  compensation.  The  solution  by  G.  H.  Darwin  for 
uncompensated  mountains  and  valleys  gave  a  maximum  stress- 
difference  equal  to  74  per  cent  of  half  the  height,  this  maximum 
occurring  at  a  depth  equal  to  about  one-sixth  the  distance  between 
mountain  crests.  Even  with  perfect  isostatic  compensation,  dis- 
tributed after  the  fashion  assumed  by  Love,  the  stress-differences 
for  mountains  and  valleys  are  seen  consequently  to  be  two-thirds 
in  value  of  those  produced  by  an  uncompensated  relief,  and  are 
approximately  one-fourth  of  the  hydrostatic  pressures.  This  frac- 
tion, one-fourth,  happens  also  to  be  the  same  as  Poisson's  ratio, 
the  ratio  of  the  lateral  expansion  to  the  vertical  shortening  of  a 
free  rock  column  under  vertical  stress. 

Now  the  distribution  of  density  has  been  found  to  be  more  or 
less  irregular,  and  there  is  no  evidence  of  such  a  reversing  layer  at 
the  base  as  Love  has  postulated.  Stress-differences  will  conse- 
quently extend  below  the  isostatic  compensation.  If,  however,  the 
latter  is  not  uniformly  distributed,  but  is  concentrated  somewhat 
in  the  outer  half  of  the  lithosphere,  the  stress-differences  will  become 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  665 

small  at  and  below  the  base  of  the  lithosphere.  On  account  of  the 
incompleteness  of  local  compensation,  the  irregularities  and  uncer- 
tainties of  the  actual  facts  of  nature,  the  Gordian  knot  of  a  solution 
may  be  cut  by  simply  assuming  for  present  purposes  the  form 
of  diagram  given  by  hydrostatic  pressures  due  to  a  compensation 
uniformly  distributed.  The  approximate  stress-differences  will  be 
given  by  taking  one-fourth  of  the  values  given  by  the  hydrostatic 
pressures.  This  transfers  the  problem  from  the  difficult  field  of 
zonal  harmonics  to  the  simple  one  of  hydrostatics,  and  perhaps  does 
not  introduce  errors  greater  than  those  involved  in  the  differences 
between  nature  and  the  postulates  which  form  the  foundation  of 
the  solution  by  zonal  harmonics.  This  hydrostatic  diagram  is 
shown  accordingly  in  Fig.  13. 

It  is  held  by  the  advocates  of  extreme  isostasy,  however,  that 
for  long-continued  stresses  the  crust  is  very  weak;  in  other  words, 
the  elastic  limit  is  low,  and  slow  plastic  deformation  readily 
occurs  which  tends  to  dissipate  the  stress-differences  and  re- 
establish isostatic  equilibrium.  To  the  extent  to  which  this 
is  true,  the  real  diagram  of  lateral  stresses  would  approach  the 
hydrostatic  diagram  here  given  and  measure  the  forces  producing 
plastic  flow. 

It  has  remained,  however,  for  the  opponents  of  the  hypothesis 
of  local  and  nearly  perfect  isostasy  to  point  out,  what  is  here 
illustrated  graphically,  that  the  extreme  theory  requires  a  belief 
in  vertical  weakness  but  lateral  strength.  If  it  were  not  for  lateral 
strength  the  land-column  would  crowd  against  the  sea-column, 
more  at  the  top  than  at  the  bottom.  Flowing  out  with  a  glacier- 
like  motion  over  the  upper  part  of  the  sea-column,  the  land-column 
would  settle  at  the  top  and  become  shorter.  This  in  turn  would 
bring  about  a  vertical  elevatory  pressure  against  its  bottom,  the 
column  would  rise,  lateral  creep  would  continue  with  equal  pace, 
and  the  end  result  would  be  a  density  stratification  in  which  the 
continental  crust  would  come  to  overlie  the  oceanic  crust.  The 
limit  of  such  an  action  would  be  given  by  the  decreasing  surface 
gradient,  this  finally  becoming  so  gentle  as  to  stop  the  glacier-like 
flow.  The  lack  of  such  an  effect  implies  of  course  that  the  lateral 
stresses  of  the  outer  part  of  the  lithosphere  lie  within  the  elastic 


666  JOSEPH  BARRELL 

limit.  Therefore  they  may  be  regarded  as  having  not  more  than 
a  quarter  of  the  value  shown  in  Fig.  i3A. 

The  suggestion  of  the  existence  of  opposing  modifying  factors  is 
to  be  found  in  conclusions  from  the  previous  parts  of  this  investi- 
gation— that  compensation  may  be  in  many  places  concentrated 
somewhat  in  the  outer  half  of  the  zone  as  here  shown  and  in  other 
places  fade  out  through  a  notable  distance  below.  These  two 
variations  in  the  distribution  of  compensation  would  modify  the 
stress  diagram  in  opposite  directions. 

Modifications  of  stresses  produced  by  base-leveling. — Consider  next 
the  case  of  complete  erosion  to  sea-level,  as  shown  in  Fig.  136. 
The  rock  from  the  land-column  has  been  deposited  as  sediment 
over  the  sea-column.  As  the  columns  are  supposed  to  act  as 
units  the  sediment  is  shown  as  spread  uniformly.  The  lateral 
stress  diagram  beneath  the  bottom  of  the  sediment  shows  a  rate 
of  decrease  the  same  as  in  case  A,  but  the  value  of  the  hydrostatic 
stress  at  any  depth  is  diminished  by  the  sum  of  the  depths  of 
erosion  and  deposition.  The  lateral  stress  now  changes  in  sign 
at  a  point  S  and  at  this  depth  is  a  line  of  no  lateral  stress.  Above 
this  depth  the  continental  segment  tends  still  to  spread  over  the 
ocean,  but  less  effectively  than  before;  below  this  depth  the  oceanic 
segment  now  thrusts  against  the  continental  crust. 

If  the  ocean  water  be  eliminated  from  the  diagram  and  base- 
leveling  should  bring  both  columns  to  a  uniform  surface,  then  the 
neutral  depth  5  advances  to  the  surface  and  the  lateral  stress 
diagram  in  B  is  just  the  reverse  in  value  to  A.  In  that  case  there 
is  no  lateral  thrust  at  the  surface,  but  at  all  depths  below  there  is 
an  excess  pressure  against  the  continent  reaching  a  maximum 
at  the  bottom  of  the  lithosphere.  This  extreme  case  cannot  apply 
to  the  ocean  except  for  that  limited  width  over  which  is  built  out 
a  continental  shelf.  To  the  degree  to  which  the  weight  of  this 
shelf  is  supported  by  the  ocean  crust  beyond,  the  column  beneath 
the  shelf  would  not  operate  with  its  full  pressures  against  the  land. 
The  case  would  apply  better  to  the  complete  erosion  of  level- 
topped  plateaus  situated  within  a  continent. 

For  the  lateral  pressure  within  the  lithosphere  to  become  effect- 
ive in  a  landward  undertow  would  require  a  lesser  rigidity  of  the 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  667 

crust  at  the  bottom  than  at  the  top.  Such  a  lesser  rigidity  may  be 
granted,  but  it  is  seen  then  that  the  landward  undertow  would  be 
greatest  at  the  bottom  and  could  not  advance  above  a  depth 
indicated  on  the  diagrams  by  T.  At  this  point  the  stress  is  of  the 
opposite  sign  but  of  the  same  value  as  for  the  state  of  isostatic 
balance  in  case  A.  If  seaward  flow  did  not  take  place  at  this  level 
in  the  first  case,  landward  flow  could  not  take  place  in  the  second. 

For  the  extreme  case  where  isostasy  is  completely  destroyed  by 
surface  leveling,  no  water  body  remaining,  T  will  rise  upward  to  a 
depth  equal  to  one-half  the  depth  of  the  zone  of  compensation. 
If  the  surface  of  complete  compensation  be  76  miles  deep,  this 
gives  a  minimum  depth  of  36  miles.  For  the  undertow  to  reach 
this  height  implies,  however,  not  only  the  limiting  case  of  complete 
destruction  of  isostasy,  but  a  crust  only  one-half  as  rigid  at  depth 
T  as  at  the  surface  and  a  previous  state  of  expansive  surface 
stress  as  great  as  the  outer  crust  could  bear.  On  the  other  hand, 
if  tangential  pressures  due  to  centrospheric  shrinkage  should 
co-operate  with  the  stresses  tending  to  restore  isostatic  equilibrium, 
underthrust  would  become  more  effective  below,  but  overthrust 
would  also  become  effective  above. 

The  disappearance  of  isostatic  compensation  at  a  certain  level 
means  the  disappearance  of  notable  heterogeneity  in  the  earth- 
shell  below,  as  argued  in  Part  V  (pp.  446-48).  One  of  the  possible 
suppositions  to  explain  this  is  to  suppose  that  this  shell  is  weaker 
than  the  crust  above  and  therefore  the  lateral  thrust  due  to  an 
assumed  initial  heterogeneity  would  cause  a  lateral  flow,  a  density 
stratification,  and  a  resulting  disappearance  of  the  postulated 
heterogeneity.  This  supposition  of  a  weaker  zone  finds  support 
in  other  lines  of  evidence.  Therefore,  although  some  lateral  flow 
at  the  base  of  the  lithosphere  may  occur  during  the  restoration  of 
isostatic  equilibrium,  it  is  to  be  expected  that  the  bulk  of  such 
flow  will  be  below,  for  there  the  substance  is  more  plastic  and  the 
lateral  stress  is  throughout  at  a  maximum. 

Let  attention  be  given  next  to  the  vertical  as  contrasted  to  the 
lateral  unbalancing  brought  in  by  the  destruction  of  isostatic 
equilibrium.  The  land-column  becomes  lighter,  the  sea-column 
heavier,  by  amounts  which  are  shown  in  the  vertically  lined  stress 


668  JOSEPH  BARRELL 

diagrams  at  the  base  of  the  lithosphere  in  case  B.  Supposing  that 
vertical  readjustment  of  the  columns  is  prevented  for  a  time  by  the 
strength  of  the  crust,  the  vertical  stresses  will  be  taken  up  by  a 
vertical  shearing  strain  along  the  partition  between  the  two 
columns.  This  shear  is  equal  in  amount  to  the  difference  in  total 
weights  of  the  columns.  Let  the  shear  per  unit  area  be  called  s. 
It  acts  over  a  surface  taken  as  122  km.  high.  Let  this  height  be 
called  h.  The  weight  of  the  columns  will  vary  with  their  breadth 
in  the  plane  of  the  drawing.  If  the  breadth  of  each  be  taken  as  b 
and  the  weights  per  unit  area  as  M  and  N  (N  including  rock,  sedi- 
ment, and  sea^  water),  then  for  a  cross-section  of  unit  thickness 
the  total  difference  in  weight  is  (N—M)b  and  the  total  shear 
is  this  same  amount,  provided  that  the  columns  are  not  sus- 
tained in  part  by  other  boundaries.  But  the  total  shear  is  also  sh. 
Therefore 

sh=(N-M)b 

s=(N-M}-h 

For  narrow  columns  b  is  small,  giving  to  5  a  small  value  and  con- 
sequently one  within  the  elastic  limit.  Let  b  become  broad  and  s 
will  then  become  large  and  exceed  the  elastic  limit.  The  lateral 
pressures,  on  the  contrary,  are  less  dependent  upon  the  breadth, 
and,  if  the  problem  were  regarded  as  one  of  hydrostatic  pressures, 
would  be  wholly  independent  of  breadth.  The  formula  shows  that 
the  broader  the  columns,  the  more  readily  they  will  readjust  by 
vertical  shear  between  the  columns.  Now  unless  failure  by  vertical 
shear  took  place  between  the  upper  part  of  the  columns  the  heavy 
column  would  be  held  up,  the  light  column  would  be  held  down, 
except  for  the  partial  effect  of  sagging  in  case  the  columns  were 
very  broad.  The  lateral  landward  pressure  at  the  base  could 
therefore  not  become  effective.  The  loaded  portion  of  the  crust 
must  fail  first  by  shear  or  flexure  of  its  upper  portion.  Whatever 
be  the  distribution  of  strength  it  would  appear  then  that  the  primary 
yielding  is  the  vertical  one  and  the  landward  force  of  undertow 
can  become  only  secondarily  effective. 

The  hypothesis  of  local  and  nearly  complete  isostasy  requires 
that  the  elastic  limit  for  vertical  shear  should  be  very  low  in  order 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  669 

that  narrow  columns  should  be  able  to  rise  or  sink.  This  may  be 
illustrated  by  the  following  example: 

Suppose  a  region  50  km.  in  radius  possesses  a  mean  departure 
from  isostatic  equilibrium  equal  to  76  m.  of  rock  (250  ft.)  and  that 
the  surrounding  regions  are  out  of  adjustment  by  the  same  amount 
but  in  the  reverse  direction.  This  is  the  maximum  area  for  regional 
isostasy  which  in  Hayford's  opinion  is  to  be  expected,  and  250  ft. 
is  the  mean  departure  from  isostasy  as  given  by  him  in  his  Minne- 
apolis address.  But  in  this  example  the  adjacent  regions  are  each 
assumed  to  be  out  of  adjustment  in  opposite  directions  by  this 
amount  and,  therefore,  the  differential  load  is  twice  this  or  500  ft. 
of  rock.  The  case  is  one  which  he  would  regard  consequently  as 
rather  extreme.  Now  a  cylinder  100  km.  in  diameter  and  122  km. 
deep  could  not  fail  through  its  bending  moment,  as  in  the  flexing 
of  a  beam.  It  would  have  to  fail  as  in  punching  a  rivet  hole 
through  a  metal  plate,  in  other  words,  by  circumferential  shear. 
The  shearing  stress  per  unit  area  is  obtained  by  dividing  the  total 
load  by  the  total  shearing  surface.  With  the  data  taken  as  above 
this  gives  s=  8 . 4  kg.  per  sq.  cm.  or  120  Ibs.  per  sq.  in.  But  strong 
rock  at  the  surface  can  readily  carry  a  shearing  stress  of  from  700 
to  1,000  kg.  per  sq.  cm.  (10,000  to  14,000  Ibs.  per  sq.  in.).  Isostatic 
perfection  to  this  degree  would  therefore  require  the  zone  of  com- 
pensation as  a  whole  to  be  only  about  one-hundredth  as  strong  under 
permanent  stress  as  is  solid  rock  at  the  surface.  This  calculation 
alone  would  tend  to  show  that  the  loads  and  areas  by  which  the 
crust  departs  from  isostatic  equilibrium  have  been  much  under- 
estimated by  the  advocates  of  extreme  isostasy. 

It  should  be  noted,  however,  that,  following  the  lines  of  his 
rejoinder  to  Lewis,  Hayford  would  answer  that  he  regarded  the 
landward  isostatic  flow  as  taking  place  within  the  zone  of  isostatic 
compensation  and  the  vertical  shear  as  operating,  consequently, 
through  a  depth  far  less  than  the  thickness  of  the  entire  zone  of 
compensation.  There  are,  however,  a  number  of  inconsistencies 
in  this  argument,  some  of  which  have  already  been  made  evident. 
Others  will  appear  as  a  result  of  the  later  discussion  of  this  chapter. 
But  it  may  be  noted  that  even  granting  this  contention — that  only 
the  outer  third  of  the  zone  of  compensation  was  involved — the 


670  JOSEPH  BARRELL 

unit  shearing  stress  would  be  multiplied  only  by  two  or  three  and 
would  still  imply  a  weakness  in  this  part  of  the  crust  to  resist 
long-enduring  shear  or  bending  stresses,  its  capacity  being  only 
3  or  5  per  cent  at  most  as  great  as  is  found  to  exist  in  surface  rocks 
for  stresses  of  human  duration. 

Relief  of  stress  accompanying  restoration  of  isostasy. — It  is  seen 
from  the  preceding  analysis  that  the  movement  of  the  unbalanced 
columns  toward  a  new  state  of  equilibrium  will  be  partly  by  vertical 
shear  in  the  neutral  ground  between  them,  but,  where  the  areas 
are  large  in  comparison  with  the  thickness  of  the  zone  of  compensa- 
tion, the  easiest  mode  of  yielding  may  be  by  flexure,  showing  at  the 
surface  as  crustal  warping.     Both  modes  of  yielding  serve  to  trans- 
mit the  excess  vertical  stresses  of  the  heavy  and  sinking  column 
into  the  asthenosphere.     If  the  latter  be  indeed  a  shell  of  weakness 
it  will  transmit  these  pressures  more  or  less  hydros tatically.     The 
vertical  pressure- differences  will  act  within  it  as  lateral  pressures 
A, '  making  for  flow  toward  the  lighter  column.     It  is  shown  in  Fig.  136 
\fV  that  the  maximum  horizontal  stress  in  so  far  as  it  approaches 
a  hydrostatic  distribution  acts  throughout  the  whole  depth  of  this 
N^  \      zone,  so  that  it  not  only  is  weaker  than  the  crust  above,  but  is 
' ;  subjected  to  maximum  stress  over  a  greater  area.     It  will  yield 

by  flowage  therefore  either  if  of  small  depth  and  very  plastic,  or  of 
great  depth  but  more  rigid.  If  the  columns  are  adjacent  and  nar- 
row as  compared  to  the  thickness  of  the  shell  of  weakness,  then  the 
principles  of  plastic  flow  would  require  that  the  flow  be  chiefly  in 
the  upper  part  of  this  shell.  If,  however,  the  columns  are  of  con- 
siderable breadth  compared  to  the  thickness  of  the  asthenosphere, 
and  especially  if  at  a  distance  from  each  other,  then  the  principle  of 
least  work  would  determine  that  the  middle  strata  of  this  shell 
should  flow  the  farthest  and  the  whole  would  to  some  degree 
participate.  If  an  imaginary  partition  were  extended  downward 
through  this  shell  as  shown  in  A  and  B  of  Fig.  13  this  partition 
would  be  found  warped  after  the  movement  as  shown  in  C  of  the 
same  figure. 

It  was  seen  in  an  earlier  part  of  this  discussion  that,  even  sup- 
posing deformation  became  effective  by  means  of  the  lateral 
stresses  within  the  lithosphere  and  without  the  existence  of  a  zone 


THE  STRENGTH  OF  THE  EARTH'S  CRUST       671 

of  weakness  below,  still  only  the  basal  part  below  the  point  T 
would  be  competent  to  give  a  landward  movement  during  the 
restoration  of  isostatic  equilibrium.  But  now  it  is  seen  that 
in  the  asthenosphere  the  lateral  pressures  are  transmitted 
with  greater  amount,  from  a  greater  distance,  and  with  a  greater 
cross-section.  The  zone  is  one  without  notable  isostatic  compen- 
sation within  it  and  is  presumably  more  plastic  than  the 
basal  part  of  the  lithosphere.  Therefore  there  is  good  reason 
to  believe  that  the  subcrustal  undertow  is  restricted  to  the 
asthenosphere. 

The  forces  actually  needed  to  produce  flowage  would  be  in 
reality  but  a  fraction  of  those  indicated  in  Fig.  136  as  existing  in 
the  asthenosphere.  The  reason  is  that  the  greater  part  of  the 
vertical  forces  is  consumed  in  producing  flexure  and  shear  in  the 
lithosphere.  Only  a  residuum  is  needed  to  produce  a  slow  plastic 
flow  in  the  shell  below.  For  that  reason  broken  lines  are  used  in 
that  part  of  the  stress  diagram.  The  energy  consumed  within 
the  lithosphere  by  its  deformation  will  be  nearly  independent  of 
the  breadth  of  the  columns;  it  will  actually  tend  to  become  some- 
what less  with  breadth  because  flexure  on  large  radii  will  be  favored. 
The  energy  consumed  in  the  asthenosphere  will,  on  the  other  hand, 
increase  with  the  breadth  of  the  columns,  but  will  be  spread  over 
a  greater  area.  The  temperature  effect  due  to  the  absorption  of 
energy  would  appear  to  be  a  minor  factor,  for  it  cannot  exceed  that 
energy  which  is  supplied  by  the  average  vertical  stress-difference 
multiplied  by  the  vertical  distance  moved.  The  average  vertical 
stress-difference  will  be  the  mean  between  that  at  the  beginning 
of  movement  and  that  residual  stress  remaining  after  the  movement 
is  completed. 

In  determining  the  scale  of  the  diagrams  of  Fig.  13  the  following 
data  were  chosen.  The  land-column  was  taken  in  A  as  having  a 
surface  elevation  of  1,000  m.  and  a  density  of  2.70;  the  sea  as 
3,000  m.  deep,  and  the  rock  below  as  possessing  a  density  of  2.77. 
The  sea- water  has  a  density  of  1.03.  These  relations  give  an 
isostatic  balance  at  a  depth  of  122  km.  In  B,  erosion  of  the  land 
to  sea-level  is  supposed  to  have  taken  place  and  the  sediment 
spread  with  same  unit  weight  over  the  sea-column  that  it  had  as 


672  JOSEPH  BARRELL 

rock  upon  the  land.     These  relations  give  a  depth  of  54  km.  to 
S  and  88  km.  to  T. 

It  should  be  repeated,  however,  in  closing  this  topic,  that  the 
solutions  here  given  are  approximate  only  and  assume  that  iso- 
static  compensation  results  in  lateral  stress-differences  which  show 
the  same  distribution  of  forces  as  a  diagram  of  hydrostatic  pressures, 
differing  only  in  magnitude.  The  writer  is  inclined  to  think  that 
the  actual  facts  of  nature  call  in  most  cases  for  some  depression  in 
depth  of  the  critical  points  beyond  those  here  shown.  Especially 
is  there  likely  to  be  under  the  margins  of  a  continent  in  isostatic 
equilibrium  some  permanent  lateral  stress-difference  within  the 
asthenosphere,  due  to  the  compensation  above  and  tending  toward 
a  landward  undertow.  Upon  the  unbalancing  due  to  erosion  and 
sedimentation  this  would  cause  the  lateral  stress-differences  within 
the  asthenosphere  to  rise  more  quickly  to  the  low  elastic  limit  and 
permit  more  readily  than  would  otherwise  be  the  case  a  regional 
readjustment  toward  isostasy. 

RELATIONS   OF   UNDERTOW  TO   THE   ZONE   OF   COMPENSATION 

Present  status  of  the  problem. — The  causes  of  vertical  movements 
Button1  made  twofold.  He  clearly  distinguished  on  the  one  hand 
between  those  internal  forces  leading  to  expansion  or  contraction, 
which  tend,  by  producing  changes  in  density,  to  create  isostatically 
a  new  surface  relief,  and,  on  the  other  hand,  those  isostatic  re- 
adjustments following  erosion  and  sedimentation,  readjustments 
which  tend  not  to  make  a  new,  but  to  restore  the  older,  relief. 
Folding  he  regarded  as  unrelated  to  the  former,  as  a  result  of  the 
latter.  He  had  shown  earlier  (in  fact,  he  had  the  honor  of  being 
the  first  to  show)  that  the  time-sanctioned  hypothesis  of  cooling 
as  a  cause  of  crustal  shrinkage  and  consequent  mountain-making 
was  inadequate  to  account  for  either  the  distribution  or  amount  of 
folding.2  From  this  he  was  led  to  regard  folding  as  due,  not  to  any 
kind  of  contraction,  but  as  a  compressive  movement  of  one  section 

1  "On  Some  of  the  Greater  Problems  of  Physical  Geology,"  Bull.  Phil.  Soc.  Wash., 
XI  (1889),  51-64- 

2  C.  E.  Button,  "A  Criticism  upon  the  Contractional  Hypothesis,"  Am.  Jour. 
Sa.,  VIII  (1874),  113-23- 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  673 

of  the  crust  against  another,  presumably  offset  by  tension  in  some 
other  region.  Button's  argument  is  that  the  crust  beneath  the 
plateau  is  unloaded  by  erosion,  that  the  crust  beneath  the  basin  is 
loaded  by  sedimentation.  An  isostatic  movement,  rejuvenating 
the  relief,  must,  by  causing  the  overloaded  basin  to  settle,  produce 
a  squeezing-out  of  matter  beneath  the  sinking  area,  and  a  crowding- 
in  of  matter  beneath  the  rising  area.  The  surficial  movement  of 
sediment  is  from  the  high  area  toward  the  low.  The  deep-seated 
movement  is  from  the  low  toward  the  high.  Thus  the  cycle 
becomes  completed  and  the  mass  of  matter  above  the  level  of  com- 
plete compensation  remains  the  same  in  each  column.  The  seaward 
movement  of  the  sediment,  as  a  factional  resistance  against  the 
river  bottoms,  produces  only  an  insignificant  drag,  but  the  return 
subterranean  movements  by  viscous  or  solid  flowage  must  produce 
a  pronounced  drag  upon  the  crust  in  the  direction  of  the  rising 
region.  Button's  reasoning  is  clear,  but  the  effectiveness  of  the 
action  rests  upon  several  assumptions.  First,  it  omits  the  influence 
of  the  surface  relief  and  the  degree  to  which  that  tends  to  a  lateral 
spreading  movement  from  the  high  toward  the  low  regions.  Sec- 
ondly, it  postulates  a  low  rigidity  to  the  crust,  as  he  in  fact  notes. 
Thirdly,  it  involves  the  conception  of  a  strong  undertow  fairly 
near  the  surface  in  order  that  the  crust  above  may  be  too  weak  to 
resist  the  viscous  drag.  As  there  were  little  quantitative  data 
available  at  the  time  when  Button  formulated  this  corollary  of  his 
theory  of  isostasy  he  could  not  have  tested  the  validity  of  these 
assumptions,  but  raised  the  problem  for  those  who  should  come 
after  him. 

This  theory  of  folding  took  a  somewhat  different  form  in  the 
mind  of  Willis,  as  expressed  in  the  concluding  chapter  of  his 
Research  in  China.1  This  work  in  many  ways  is  of  the  very  first 
importance  and  gives  a  comprehensive  view  of  the  geological  history 
of  the  whole  continent  of  Asia.  As  to  the  nature  of  the  movements, 
he  finds  that  the  continent  of  Asia  may  be  resolved  into  positive 
and  negative  elements,  the  former  areas  tending  to  stand  high,  the 
latter  tending  to  stand  low.  These  tendencies  are  latent  during 
comparatively  long  periods  of  quiet  and  resultant  peneplanation, 

1  Vol.  II  (1907),  Carnegie  Institution  of  Washington. 


674  JOSEPH  BARRELL 

but  become  operative  during  epochs  of  diastrophism.  The  com- 
pressive  movements,  on  the  other  hand,  have  pressed  and  welded 
the  positive  elements  together,  the  axial  directions  of  folding 
representing  the  compression  of  the  negative  zones  lying  between. 

The  cause  of  the  diastrophism  Willis  ascribes  to  differences  in 
specific  gravity,  restricted,  according  to  Hayford's  determination, 
to  the  outer  hundred  miles  of  the  earth's  body ;  the  vertical  move- 
ments being  chiefly  due  to  isostatic  readjustment  between  the 
several  continental  elements,  the  compressive  movements  being 
due  to  the  tendency  of  the  heavier  oceanic  segments  of  the  earth 
to  spread  and  underthrust  the  outer  portions  of  the  whole  conti- 
nental mass.  This  theory  of  the  cause  of  lateral  compression  was 
discussed  by  the  present  writer  in  a  review  of  Willis'  work,1  and 
the  objections  stated  against  it  there  are  in  part  the  same  as  will 
be  elaborated  farther  on  in  the  present  article. 

Hayford  took  up  the  same  subject  in  his  address,  delivered  at 
Minneapolis  on  December  29,  1910,  as  retiring  vice-president  of 
Section  D  (Mechanical  Science  and  Engineering)  of  the  American 
Association  for  the  Advancement  of  Science,  the  title  of  his  paper 
being  "The  Relations  of  Isostasy  to  Geodesy,  Geophysics,  and 
Geology."2  This  is  a  paper  of  broad  scope  intended  to  show  how 
vertical  movements  not  in  apparent  accord  with  isostasy  and  also 
movements  of  folding  may  be  explained  as  secondary  results  of 
isostatic  adjustment  and  really  in  harmony  with  the  hypothesis 
of  nearly  continuous  movement  in  a  crust  of  low  rigidity  and  of 
almost  complete  isostasy.  This  part  of  his  theory  is  essentially 
the  same  as  Button's  but  is  elaborated  in  greater  detail. 

Harmon  Lewis  called  attention  to  the  defects  in  this  theory  of 
deformation,3  but  Hayford  made  a  rejoinder,  positive  and  sweeping 
in  its  style,  to  this  and  other  lines  of  criticism  by  Lewis.4 

The  names  of  Button,  Willis,  and  Hayford  deservedly  carry 
much  weight  and  must  be  accepted  at  their  face  value  by  geologists 

'  Science,  N.S.,  XXIX  (1909),  257-60. 

2  Published  in  Science,  N.S.,  XXXIII  (1911),  199-208. 

3  "The  Theory  of  Isostasy,"  Jour.  Geol.,  XIX  (1911),  620-23. 

4  John  F.  Hayford,  "Isostasy,  a  Rejoinder  to  the  Article  by  Harmon  Lewis," 
Jour.  Geol.,  XX  (1912),  562-78. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  675 

who  have  not  themselves  made  a  critical  study  of  the  problems  of 
isostasy.  The  arguments  which  the  writer  advanced  in  1909 
against  this  hypothesis  were  published  under  a  title  which  appar- 
ently did  not  call  attention  to  them.  The  style  of  Hayford's  reply 
to  Lewis  is  crushing  and  conveys  the  impression  that  Lewis  has  been 
completely  refuted.  It  is  because  of  these  reasons  that  the  sub- 
ject calls  here  for  fuller  development. 

In  his  Minneapolis  address  Hayford  outlines  a  theory  of  the 
principles  of  diastrophism  which  turns  upon  his  conclusion  that 
isostasy  is  so  nearly  complete  that  areas  of  even  limited  size  average 
only  250  feet  from  the  level  of  isostatic  equilibrium.  He  assumes 
chemical  and  physical  changes  to  be  induced  in  the  crust  by 
the  changing  load  due  to  erosion  and  sedimentation.  These  he 
thinks  are  superimposed  upon  the  effects  of  nearly  continuous 
vertical  movements  of  isostatic  readjustment.  The  vertical  move- 
ments in  turn  produce  a  lateral  undertow  which  is  given  as  a  cause 
of  localized  heating  and  folding.  Apparently  this  is  regarded  as 
a  complete  mechanism  of  deformation  since  the  author  raises  the 
query : 

Is  it  at  all  certain  that  under  the  influence  of  such  actions  the  geological 
record  at  the  earth's  surface  at  the  end  of  fifty  to  one  hundred  million  years 
would  be  appreciably  less  complicated  than  the  geologic  record  which  is  actually 
before  us  ?  I  think  that  it  would  be  fully  as  complicated  as  the  actual  record.1 

This  theory  of  folding  as  the  result  of  subcrustal  undertow  is 
illustrated  by  means  of  two  diagrams.  In  Fig.  i ,  the  zone  of  viscous 
flow  from  the  sinking  toward  the  rising  area  is  placed  in  the  lower 
quarter  of  the  zone  of  isostatic  compensation.  In  Fig.  2  it  is  shown 
in  the  middle  of  that  zone,  dying  out  both  above  and  below. 
Apparently  then,  as  shown  by  these  two  different  conceptions,  the 
author  cited  was  guided  by  no  definite  theory,  based  upon  the 
mechanics  of  materials,  as  to  the  factors  which  would  determine 
the  depth  of  this  zone  of  undertow  and  its  relations  to  the  zone  of 
compensation. 

Harmon  Lewis  in  his  paper  on  the  "Theory  of  Isostasy"  has 
discussed  various  aspects  of  the  isostatic  theory  as  developed  by 

1  Op.  tit.,  p.  206. 


676  JOSEPH  BARRELL 

Hayford,  and  among  them  this  question.     Regarding  the  possi- 
bility of  folding  by  means  of  isostatic  undertow,  Lewis  concludes: 

Now,  according  to  the  theory  of  isostasy,  compensation  would  be  essen- 
tially complete,  and  if  compensation  is  complete  the  depth  of  compensation 
as  determined  by  Hayford's  geodetic  work  would  be  as  great  as  60  miles. 
Hence,  the  undertow  postulated  by  isostasy  would  exist  chiefly  below  60  miles. 
It  is  decidedly  questionable  that  an  undertow  even  much  nearer  to  the  surface 
than  60  miles  would  cause  the  observed  folding  in  the  upper  few  miles  of  the 
crust.1 

In  regard  to  this  criticism  by  Lewis  concerning  the  cause  of 
folding,  Hayford  states  in  reply: 

On  pp.  621-22  Mr.  Lewis  sets  forth  the  argument  that  there  is  much 
geological  evidence  of  horizontal  movements  in  the  outside  portions  of  the 
earth,  especially  in  the  form  of  folding,  that  the  controlling  movements  of 
isostasy  are  assumed  to  be  vertical  and  hence  cannot  account  for  folding,  and 
that  the  horizontal  movement  or  undertow  concerned  in  isostatic  readjustment 
must  be  below  the  depth  of  compensation  and  hence  so  far  below  the  surface 
as  to  be  very  ineffective  in  producing  folding. 

There  are  two  fatal  defects  in  this  argument  as  applied  to  controverting 
anything  that  Hayford  believes  or  has  written. 

First,  Hayford  has  already  indicated  clearly  his  belief  that  the  undertow 
concerned  in  isostatic  readjustment  is  above,  not  below,  the  depth  of  compensa- 
tion. In  both  the  figures  published  in  his  Minneapolis  address  the  undertow 
is  clearly  indicated  as  being  above  the  depth  of  compensation  and  it  is  also 
so  indicated  in  the  corresponding  text.  As  Hayford  puts  the  undertow  com- 
paratively near  the  surface,  where  it  is  conceded  that  it  would  be  effective  in 
producing  folding,  the  existence  of  extensive  folding  is  a  confirmation,  not  a 
contradiction,  of  his  theory  of  the  manner  in  which  isostatic  readjustment 
takes  place.  It  is  certainly  not  fair  to  hold  Hayford  responsible,  either  directly 
or  by  inference,  for  any  theory  which  someone  else  may  believe  which  involves 
an  undertow  situated  entirely  below  the  depth  of  compensation.  Mr.  Lewis 
apparently  believes  such  a  theory. 

Second,  the  movements  which  produce  isostatic  readjustment  are  neces- 
sarily horizontal,  not  vertical.  If  two  adjacent  columns  of  the  same  horizontal 
cross-section  extending  from  the  surface  to  the  depth  of  compensation  have 
different  masses  the  readjustment  to  perfect  compensation  must  involve  a 
transfer  of  mass  out  of  one  column,  or  into  the  other,  or  from  one  to  the  other. 
In  any  case  the  transfer  must  be  a  horizontal  movement.  Hayford  has  already 
shown  in  print  more  than  once  that  he  understands  that  vertical  movement 
alone  does  not  produce  isostatic  readjustment.  Moreover,  a  careful  reading 

lOp.  cit.,  p.  622. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  677 

of  his  Minneapolis  address  will  certainly  show  that  he  believes  that  the  total 
amount  of  material  moved  horizontally  during  isostatic  readjustment,  and 
especially  the  total  number  of  ton-miles  of  such  movement,  is  vastly  in  excess 
of  the  corresponding  quantities  concerned  in  the  vertical  components  of  the 
movement  which  takes  place.  Hence  the  folding  and  other  abundant  evidence 
of  past  horizontal  movements  observed  by  geologists  confirm  Hayford's 
hypothesis  as  to  the  manner  in  which  isostatic  readjustment  takes  place, 
instead  of  conflicting  with  it  as  Mr.  Lewis'  article  would  lead  one  to  think.1 

The  present  writer,  however,  believes  with  Mr.  Lewis  in  the 
theory  that  an  undertow  must  be  essentially  below  the  zone  of 
compensation  and  is  incapable  of  producing  surficial  folding.  The 
reasons  have  been  given  in  part  in  the  consideration  of  the  stress- 
relations,  as  they  would  exist  under  the  hypothesis  of  extreme 
isostasy.  But  there  are  other  reasons  why  the  subject  should  be 
discussed  in  further  detail.  One  reason  is  that,  if  Lewis  is  right 
on  this  point  and  Hayford  wrong,  it  is  desirable  that  this  should 
be  made  clear,  in  justice  to  Mr.  Lewis  as  well  as  to  the  subject. 
The  other  reason  is  that  here  in  reaching  a  conclusion  we  can 
advantageously  pursue  a  method  of  exclusion.  By  showing  that 
isostatic  undertow  cannot  take  place  within  the  zone  of  compen- 
sation, for  various  reasons  besides  those  discussed  in  the  stress 
diagrams,  we  reach  the  conclusion  that  it  must  take  place  in  a 
level  below  that  zone.  Furthermore,  by  noting  the  conditions 
which  would  hinder  lateral  flowage  we  may  arrive  at  a  conclusion 
as  to  those  which  must  exist  to  greater  or  less  degree  in  order  to 
permit  it. 

Objections  against  undertow  in  the  zone  of  compensation. — The 
pressures  which  occur  during  a  state  of  isostasy  and  after  the 
destruction  of  that  condition  have  been  discussed.  It  was  seen 
that  the  pressures  making  for  the  undertow  necessary  to  restore 
isostasy  were  greatest  at  the  bottom,  but,  more  especially,  below 
the  bottom  of  the  zone  of  compensation.  The  possibility  remains 
to  be  considered,  however,  that  perhaps  the  distribution  of  the 
rigidity  of  the  crust  more  than  offsets  the  distribution  of  pressures. 
Suppose  the  middle  of  the  zone  of  compensation  should  be  very 
weak  and  the  crust  at  and  below  the  bottom  be  very  strong.  Then, 

1  Op.  cit.,  pp.  573,  574. 


678  JOSEPH  BARRELL 

if  the  restoration  of  isostasy  was  deferred  until  assisted  by  strong 
tangential  pressures  due  to  centrospheric  shrinkage,  it  might  be 
held  that  isostatic  undertow  could  take  place  within  the  zone  of 
compensation  and  between  5  and  T  of  Fig.  136.  If,  furthermore, 
compensation  should  be  not  uniformly  distributed  but  taken  as 
largely  concentrated  in  the  upper  part  of  the  zone  of  compensation, 
which  however  is  contrary  to  the  Hayfordian  hypothesis,  then  the 
forces  making  for  undertow  may  correspondingly  rise  in  the  crust. 
For  these  reasons  it  is  seen  that  the  previous  argument  from  the 
distribution  of  pressures  is  not  final  and  that  the  physical  conditions 
involved  in  lateral  flowage  must  also  be  considered. 

The  only  positive  reason  which  has  been  advanced  for  seeking 
to  place  the  undertow  within  the  zone  of  compensation  is  in  order 
to  utilize  its  viscous  drag  as  a  cause  of  folding.  To  become  effective 
the  drag  must  be  strong,  the  crust  above  by  contrast  weak  and 
therefore  thin.  The  crumpling  pressure  on  the  surface  of  the 
crust  cannot  be  transmitted  directly  from  the  sinking  area,  as  is 
shown  in  Fig.  13,  since  the  thrusting  force  is  greatest  at  the  bottom. 
It  must  be  supposed  to  arise  from  the  viscous  drag  of  the  undertow. 
But  viscosity  decreases  the  hydrostatic  head  with  increasing  dis- 
tance from  the  source.  Therefore,  to  permit  a  viscous  flow  at 
a  distance  from  the  source  of  pressure  implies  a  mobility  within 
that  level  of  the  crust  which  would  make  it  wholly  incapable  of 
carrying  the  stresses  necessary  to  maintain  its  own  isostatic 
equilibrium.  Therefore  this  level,  by  the  very  terms  of  the  general 
conception  of  isostasy,  would  become  the  bottom  of  the  zone  of 
compensation. 

As  another  mechanical  defect  of  the  theory  under  review,  it 
is  to  be  noted  that  the  section  of  undertow  taken  by  Hayford  as 
in  the  middle  of  the  zone  of  compensation  is  not  given  as  involving 
more  than  half  of  that  zone.  This  is  as  if  a  viscous  fluid  were 
transmitted  through  a  pipe  in  which  the  cross-section  of  pipe  and 
fluid  were  equal.  To  assume  that  the  fluid  is  free  to  escape  into 
a  region  of  less  pressure  at  the  far  end  and  yet  gives  such  a  fric- 
tional  resistance  against  its  walls  as  to  be  able  to  crumple  up  the 
pipe  is  to  assume  that  the  two  are  of  the  same  order  of  strength. 
The  materials  of  pipe  and  fluid  might  almost  be  interchanged. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  679 

In  such  viscous  flow  the  tendency  would  be  for  a  swelling  and  burst- 
ing to  appear  at  the  near  end  rather  than  a  through  flowage  with 
a  crumpling  of  the  pipe  at  the  far  end. 

Finally,  the  greatest  theoretical  difficulty  is  encountered  when 
it  is  sought  to  transmit  matter  from  beneath  the  regions  of  oceanic 
marginal  sedimentation  to  beneath  the  regions  of  a  continental 
interior.  Either  directly  or  indirectly  there  must  be  a  subcrustal 
transference  going  forward  all  the  way  between  these  distant 
regions;  for  example,  from  beneath  the  Mississippi  and  Colorado 
deltas  to  the  fields  of  erosion  in  the  Rocky  Mountains,  if  a  condi- 
tion of  even  approximate  isostasy  is  to  be  maintained  throughout. 
This  does  not  mean  of  course  that  an  individual  ton  of  plastic 
rock  is  transferred  a  thousand  miles  to  balance  a  ton  of  sediment. 
Each  subcrustal  unit  may  be  transferred  only  a  mile,  but  it  involves 
a  subsurface  movement  of  matter  all  the  way  from  the  regions  of 
sedimentation  to  the  regions  of  erosion. 

Now  this  implies  a  continuous  pressure-gradient,  and  even  under 
the  conception  of  great  crustal  weakness,  a  pressure-gradient  which 
could  fold  the  weak  cover-rocks  would  be  far  higher  than  that 
needed  for  the  movement  of  a  continental  glacier.  Any  large 
degree  of  viscous  resistance  in  the  zone  of  undertow  would  there- 
fore require,  in  order  to  initiate  movement,  an  enormous  defect 
of  isostasy  under  the  distant  continental  interior,  an  enormous 
excess  under  the  marginal  oceans.  After  a  rejuvenative  movement 
had  started,  it  would  be  slow,  the  frictional  and  deformative  resist- 
ances nearly  balancing  the  deforming  force.  Therefore  inertia  of 
the  moving  mass  could  not  carry  it  appreciably  beyond  the  point 
where  the  moving  force,  weakened  by  loss  of  head,  would  just 
balance  the  resistances  to  further  movement.  It  would  be  ex- 
pected, in  consequence,  that  a  residual  pressure-difference  would 
remain,  even  after  a  period  of  restorative  isostatic  movement. 
But  an  inspection  of  the  map  of  New  Method  anomalies  given  in 
Part  II,  p.  153,  does  not  show  any  such  anomaly  gradients  as  would 
comport  with  this  expectation.  A  vast  region  of  the  continental 
interior  extending  from  Lake  Superior  to  the  Rio  Grande  and  west- 
ward to  beyond  the  front  ranges  of  the  Rocky  Mountains  shows 
average  positive  anomalies,  indicating  an  excess  of  matter,  not  a 


68o  JOSEPH  BARRELL 

deficiency.  To  the  westward  is  a  broad  region  of  average  negative 
anomaly  reaching  a  maximum  at  centers  near  the  Pacific  coast 
and  no  marked  excess  is  shown  near  the  mouths  of  the  great 
rivers.  Such  a  lack  of  regional  relations  would  appear  to  show  that 
the  anomalies  are  due  much  more  to  local  loads  and  irregularities 
upon  and  within  the  lithosphere,  and  to  bowings  due  to  great 
compressive  movements  unrelated  to  isostasy,  rather  than  to  the 
existence  of  an  isostatic  gradient  leading  from  the  ocean  borders 
to  the  interior  fields  of  great  erosion.  Therefore  either  the  idea 
of  strong  viscous  drag  by  undertow  or  else  the  very  doctrine  of 
isostasy — one  or  the  other — must  be  abandoned.  But  it  has  been 
seen  that  if  undertow  exists  in  a  comparatively  plastic  stratum,  then 
that  physical  condition  will  cause  it  to  be  the  bottom  of  the  zone 
of  compensation.  Thus  the  application  of  every  pertinent  engineer- 
ing principle  reduces  the  initial  hypothesis  of  surface  folding  by 
isostatic  undertow,  and,  especially  by  undertow  within  the  zone  of 
compensation,  to  an  absurdity. 

Undertow  restricted  to  a  sphere  of  weakness — the  astheno  sphere  — 
All  of  this  accumulative  argument  has  not  been  advanced  merely 
to  show  that  a  certain  view  is  wrong.  Rather  has  it  been  the  inten- 
tion to  prepare  the  ground  for  what  would  appear  to  be  a  sounder 
theory  of  the  mode  of  maintenance  of  isostatic  equilibrium. 

As  for  the  basis  of  that  theory,  Schweydar,  from  the  mathe- 
matical analysis  of  the  measurement  of  the  tides  in  the  crust  by 
means  of  the  horizontal  pendulum,  has  found  that  they  are  in 
accord  with  the  assumption  of  the  existence  of  a  slightly  plastic 
zone  about  600  km.  thick  beneath  a  more  rigid  crust  120  km.  thick.1 
It  would  appear  that  the  geodetic  evidence  of  isostasy  points 
also  toward  the  existence  of  such  a  thick  and  somewhat  plastic 
zone  beneath  the  more  rigid  lithosphere.  It  gives  no  knowledge 
of  the  exact  thickness  or  depth,  but  for  convenience  the  figures 
given  by  Schweydar  will  be  assumed.  It  is  a  matter  of  importance 
to  note  however  that,  although  the  quantitative  limits  are  uncer- 
tain, the  suggestions  given  both  by  the  tides  and  by  isostatic 

1  "Untersuchungen  iiber  die  Gezeiten  der  festen  Erde  und  die  hypothetische 
Magmaschicht,"  Verojfentlichung  des  k.  k.  Preusz.  geodat.  Institutes,  Neue  Folge  No.  54, 
Leipzig  (1912,  B.  G.  Teubner). 


THE  STRENGTH  OF  THE  EARTH'S  CRUST 


68 1 


compensation  point  to  a  zone  of  weakness  much  deeper  and  thicker 
than  the  figures  which  have  customarily  been  taken  as  a  probable 
depth  of  origin  of  magmas.  The  latter  however  rests  upon  uncer- 
tain extrapolation,  whereas  the  figures  for  the  limits  of  the  astheno- 
sphere,  although  of  no  exactness  and  perhaps  20  or  50  per  cent 
from  limits  which  finally  may  be  chosen,  have  at  least  been  deter- 
mined by  more  direct  evidence.  In  such  a  thick  shell  of  weakness, 
the  readjustment,  after  an  erosion  cycle,  of  a  continental  interior 
to  isostatic  equilibrium  would  require  but  very  little  viscous  shear 
and  but  little  lateral  movement. 


Cenlrosp/iere 

FIG.  14. — Diagrammatic  vertical  section  of  the  crust,  to  show  nature  of  undertow 
in  the  asthenosphere  necessary  to  restore  isostatic  equilibrium  in  a  positive  interior 
continental  area  after  a  cycle  of  erosion.  Effects  of  a  vertical  movement  of  o .  5  km. 
exaggerated  60  times.  Asthenosphere  grades  into  contiguous  spheres  and  best  limita- 
tions in  depth  are  not  known. 

To  give  quantitative  visualization  to  this  conclusion  Fig.  14  is 
drawn.  Suppose  a  plateau  area  i,oookm.  wide  in  a  continental 
interior  to  be  separated  from  the  region  of  sedimentary  deposit 
by  an  intermediate  region  i,oookm.  across.  Take  a  section 
i  km.  wide  through  these  regions.  Let  an  erosion  cycle  cause  the 
removal  on  the  average  of  0.5  km.  of  rock  from  this  area  to  be 
deposited  over  an  equal  area  of  sea-bottom.  Then,  during  an 
epoch  of  diastrophism,  assume  complete  recovery  of  isostatic 
equilibrium  by  undertow  in  a  sublithospheric  'zone  of  weakness 
600  km.  thick.  The  vertical  section  of  rock  eroded  is  500  sq.  km. 
in  area.  As  we  have  chosen  a  width  of  section  of  i  km.  we  may 
also  speak  of  this  as  the  volume,  500  cu.  km.  To  restore  the  mass 
of  this  column,  500  cu.  km.  must  be  added  to  it  and  flow  past  the 
vertical  line  which  bounds  it  on  the  seaward  side.  As  this  zone  of 


682  JOSEPH  BARRELL 

flow  is  600  km.  deep,  the  actual  lateral  movement,  if  all  depths 
move  equally,  will  be  but  0.83  km.,  since  0.83X600  =  500.  If  the 
flowage  is  supposed  to  increase  regularly  from  top  and  bottom  to 
the  middle  the  movement  of  the  middle  layer  would  be  i .  66  km. 
A  previously  vertical  line  600  miles  long  through  this  asthenosphere 
would  then  be  bent  at  the  middle  by  this  amount  and  its  two 
halves  make  angles  of  o°igf  with  the  vertical.  Each  layer  a  kilo- 
meter thick  would  move  horizontally  5.6m.  with  respect  to  each 
adjacent  layer  of  kilometer  thickness.  These  figures  bring  out  the 
insignificant  degree  of  the  plastic  deformation  in  such  a  deep  zone 
which  is  needed  to  restore  isostatic  equilibrium,  even  for  a  large 
interior  continental  area  after  erosion  amounting  to  two-thirds 
of  the  present  average  elevation  of  the  North  American  continent. 

As  a  matter  of  fact  the  cross-section  of  the  plastic  deformation 
would  not  be  a  triangle,  but  a  sinusoidal  curve,  so  that  the  maximum 
linear  flow  for  thickness  of  600  km.  would  be  between  0.83  and 
1.66  km. 

This  illustration  makes  it  clear  that  the  isostatic  rejuvenation 
of  continental  interiors  as  well  as  of  the  margins,  which  meets  such 
grave  difficulties  under  the  hypothesis  of  a  thin  and  shallow  zone 
of  isostatic  undertow,  is  eliminated  by  adopting  the  hypothesis 
of  a  thick  and  plastic  sublithospheric  shell,  such  as  has  been  found 
to  be  suggested  by  independent  evidence. 

The  idea  of  folding  as  a  result  of  isostatic  undertow  definitely 
may  be  abandoned,  but  the  absence  of  a  notable  isostatic  gradient 
has  some  further  significance.  It  is  seen  from  Fig.  14  that  if  the 
fields  of  great  erosion  and  deposition  are  within  a  few  hundred 
kilometers  of  each  other  the  rejuvenative  undertow,  under  the 
laws  of  stress  distribution  in  plastic  bodies,  would  involve  mostly 
a  limited  tract  in  the  outer  part  of  the  asthenosphere;  whereas,  if 
the  undertow  must  extend  over  distances  of  i,oookm.  or  more, 
then  the  whole  depth  of  the  asthenosphere  will  become  involved. 
The  amount  of  stress-difference  and  of  plastic  shear  per  unit  of 
volume  may  therefore  be  no  greater  in  the  one  case  than  in  the  other. 
Especially,  if  the  middle  of  the  asthenosphere  is  its  weakest  part, 
a  movement  generated  by  areas  large  enough  to  involve  the  whole 
of  this  zone  would  go  forward  under  less  stress-difference  per  unit 


THE  STRENGTH  OF  THE  EARTH'S  CRUST       683 

of  area  than  for  more  local  adjustments.  The  absence  of  a  notable 
continental  gradient  is  suggestive  therefore  of  a  deep  zone  of  weak- 
ness, least  resisting  in  its  central  portions,  and  of  very  marked 
plasticity  in  comparison  with  the  rigidity  of  the  lithosphere  above. 
This  does  not  involve,  however,  the  conception  of  a  truly  fluid 
zone,  but  merely  that  of  a  comparatively  plastic  solid. 

The  existence  and  nature  of  this  zone  of  weakness  is  seen  to 
enter  vitally  into  the  theory  of  isostasy  and  must  of  course  bear 
with  equal  importance  on  other  branches  of  terrestrial  dynamics 
as  well.  It  is  proposed  therefore  to  elevate  it  to  equal  rank  with 
the  other  shells  of  the  earth  and  to  name  it  for  that  quality  which, 
from  the  standpoint  of  diastrophism,  is  its  most  significant  feature 
as  compared  to  the  zones  above  and  below.  This  is  its  inability 
to  resist  stress-differences  above  a  certain  small  limit.  Its  name, 
therefore,  is  the  sphere  of  weakness — the  asthenosphere. 

[To  be  continued] 


VOLUME  XXII  NUMBER  8 

I 

REPRINTED  FROM 
THE 

JOURNAL   OF    GEOLOGY 

NOVEMBER-DECEMBER 


THE  STRENGTH  OF  THE  EARTH'S  CRUST 


JOSEPH  BARRELL 
New  Haven,  Connecticut 


PART  VII.    VARIATION  OF  STRENGTH  WITH  DEPTH  AS  SHOWN 
BY  THE  NATURE  OF  DEPARTURES  FROM  ISOSTASY 

INTRODUCTION  AND  SUMMARY    ...  •     729 

SECTION  A,  PRESENTATION  OF  THEORY1 

RELATIONS  OF  LOADS  AND  STRESSES 73 2 

Stresses  Imposed  by  Harmonic  Surface  Loads  .  .  .  -732 
Modifications  Imposed  by  Long  and  Large  Wave-Lengths  .  .  736 
Nature  of  Stresses  Imposed  by  Internal  Loads  ....  737 
Nature  of  Stresses  Imposed  by  Perfect  Isostasy  .  .  ...  740 

INTRODUCTION  AND    SUMMARY 

The  first  five  parts  of  this  article  have  concurred  in  showing 
that  the  crust  is  very  strong  when  measured  by  its  capacity  to 
support  great  deltas,  individual  mountain  ranges,  or  great  internal 
loads  due  to  irregularities  in  density  not  in  accord  with  the  topog- 
raphy. On  the  other  hand,  the  altitudes  of  the  continents  as  a 
whole,  or  of  large  sections  of  the  continents,  agree  with  the  demands 
of  nearly  perfect  isostasy.  In  Part  VI  it  was  shown,  however, 
that,  although  even  perfect  isostasy  threw  very  considerable  stresses 

Section  B,  on  Applications  of  the  Theory,  will  be  published  in  the  folio  whig 
number  of  this  Journal.     The  Introduction  and  Summary  apply  however  to  both 
sections. 
Vol.  XXII,  No.  8  729 


730  JOSEPH  BARRELL 

upon  the  outer  part  of  the  crust,  the  maintenance  or  restoration  of 
the  isostatic  condition  through  geologic  time  in  spite  of  the  opposing 
geologic  activities  implied  the  existence  of  an  undertow  below  the 
zone  of  compensation.  The  existence  of  this  regional  isostasy 
for  continental  interiors  as  well  as  for  ocean  basins  suggests,  fur- 
thermore, that  this  zone  of  undertow  is  both  thick  and  relatively 
very  weak  to  resist  shearing  stresses.  But  if  such  a  zone  exists 
it  must  have  important  bearings  on  other  branches  of  terrestrial 
dynamics  besides  that  of  isostasy.  Its  importance  gives  it  a  right 
to  a  distinct  name,  and  it  has  been  called  here  the  zone  of  weakness — 
the  asthenosphere.  It  is  desirable  to  test  its  reality  and  its  char- 
acter by  other  lines  of  evidence,  and  such  another  line  forms  the 
basis  of  this  part. 

George  H.  Darwin  has  investigated  the  problem  of  the  stress- 
differences  imposed  on  the  earth  by  the  weights  of  continents  and 
mountains.  In  his  work  the  earth  was  assumed  to  possess  com- 
petent elasticity  throughout  and  the  topography  to  be  without 
isostatic  compensation.  Love  has  more  recently  treated  the 
contrary  problem  of  the  isostatic  support  of  continents  and  moun- 
tains, assuming  as  governing  conditions  that  isostatic  compensa- 
tion was  perfect  within  a  depth  of  one-fiftieth  of  the  earth's  radius, 
127  km.,  and  that  all  shearing  stresses  due  to  topography  and 
compensation  disappeared  at  that  depth.  Below  there  is  assumed 
to  exist  only  hydrostatic  pressures.  In  other  words,  Darwin  pos- 
tulates no  isostasy  and  no  asthenosphere ;  Love  postulates  perfect 
isostasy  and  a  perfect  asthenosphere.  As  there  is  known  to  be  no 
truly  fluid  universal  shell  within  the  earth,  and  as  isostasy  for 
limited  regions  is  far  from  perfect,  the  truth  must  lie  between  these 
two  extremes.  The  asthenosphere  must  have  some  degree  of 
strength  and  a  measure  of  its  strength  is  derived  in  this  part  by  a 
study  of  the  nature  of  the  departures  from  isostasy. 

For  this  purpose  is  discussed  the  nature  of  the  stresses  as  worked 
out  by  Darwin.  Then  the  departures  from  isostasy  are  analyzed 
into  harmonic  series.  Those  of  long  wave-length  are  seen  to  be 
of  low  amplitude,  those  of  short  wave-length  of  high  amplitude. 
Now  the  departures  from  isostasy  are  according  to  their  very  na- 
ture without  compensation  and  their  stress  effects  will  therefore 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  731 

follow  Darwin's  law  except  in  so  far  as  the  great  rigidity  of  the 
outer  crust  will  permit  it  to  sustain  loads  after  the  manner  of  a 
continuous  beam.  It  is  shown,  however,  that  the  outer  crust  is  in- 
efficient as  a  beam,  so  that  the  results  of  applying  Darwin's  analysis 
will  probably  not  be  greatly  modified. 

It  is  found  that  the  departures  from  isostasy  are  such  as  throw 
great  stress-differences  upon  the  zone  of  compensation,  here  called 
the  lithosphere.  The  maximum  stresses  imposed  by  the  loads 
found  to  exist  within  the  United  States  lie  furthermore  within 
the  outer  two-thirds  of  that  zone.  The  stress-differences  due  to 
this  cause  reach  maxima  probably  between  3,000  and  5,000  pounds 
per  square  inch. 

Within  the  asthenosphere,  on  the  contrary,  the  stresses  caused 
by  the  departures  from  isostasy  are  very  small,  under  the  United 
States  the  stress-differences  at  depths  of  from  400  to  600  km. 
reaching  maxima  probably  between  500  and  600  pounds  per  square 
inch,  between  a  sixth  and  tenth  of  those  existing  at  higher  levels. 

Now  the  nature  of  those  geologic  actions  which  oppose  isostasy, 
both  the  great  compressive  movements  and  the  great  cycles  of 
erosion  and  sedimentation,  are  such  that  they  tend  to  destroy  the 
isostatic  adjustments  of  whole  continents  and  large  parts  of  conti- 
nents. By  these  broad  actions  they  tend  to  bring  larger  and  larger 
stress-differences  upon  the  zone  lying  more  than  200  km.  deep. 
The  limitation  of  their  action  as  shown  by  the  dominance  of  regional 
isostasy  is  therefore  to  be  regarded  as  an  effect  of  weakness  in  that 
zone.  This  then  is  another  proof  of  the  reality  of  an  astheno- 
sphere. The  proof  in  Part  VI  depended  upon  the  dynamics 
necessary  for  isostatic  undertow;  the  proof  in  this  part  depends 
upon  the  limitations  of  stress  with  depth  as  measured  by  the 
existing  departures  from  isostasy. 

This  is  as  far  as  the  present  fragmentary  data  and  imperfect 
theory  can  safely  go,  but  in  order  to  visualize  the  arguments  a 
curve  of  strength  is  given  which  shows  how  great  a  falling-off  of 
strength  there  is  from  the  upper  part  of  the  lithosphere  to  the  middle 
of  the  asthenosphere.  Below,  the  strength  undoubtedly  again  in- 
creases, but  the  evidence  for  that  is  supplied  by  other  lines  of 
research  than  that  opened  by  the  geodetic  data. 


732  JOSEPH  BARRELL 

The  results  of  this  part  suggest  that  in  future  investigations  by 
mathematicians  upon  the  elastic  competence  of  the  earth,  a  prob- 
able case  would  be  to  consider  the  isostatic  compensation  as  uni- 
formly tapering  out  through  a  depth  twice  as  great  as  the  depth 
given  for  uniformly  distributed  compensation,  that  is,  tapering  out 
through  about  244-254  km.  Further,  it  is  not  in  accordance  with 
nature  to  assume  that  at  this  depth  all  shearing  stresses  disappear. 
Such  an  assumption  brings  in  artificialities  nearly  as  great  as  those 
involved  in  Darwin's  assumption  of  no  isostatic  compensation. 
Rather  should  the  stress  relations  be  solved  as  limited  by  some  such 
curve  as  is  here  shown,  and  determined  by  the  nature  of  the  depar- 
tures from  isostasy.  It  is  possible  that  this  may  add  still  further 
difficulties  to  the  mathematical  treatment  of  the  subject,  yet  only 
by  closer  recognition  of  the  realities  of  nature  can  mathematical 
analysis  become  of  increasing  value. 

SECTION  A,  PRESENTATION  OF  THEORY 
RELATIONS    OF   LOADS   AND    STRESSES 

Stresses  imposed  by  harmonic  surface  loads. — A  harmonic  series 
gives  a  succession  of  sweeping  curves  such  as  are  shown  in  Fig.  15. 
The  vertical  scale  may  be  made  of  any  size  and  the  curves  may 
be  regarded  as  sections  across  a  series  of  hills  and  valleys,  or,  on 
progressively  larger  scales,  anticlinoria  and  synclinoria,  geanticlines 
and  geosynclines,  continents  and  ocean  basins.  The  parts  of  the 
curves  convex  upward  will  then  represent  loads  above  the  mean 
surface  and  give  rise  to  stresses  acting  downward.  The  broad 
hollows  give  negative  loads  and  the  surface  beneath  is  strained 
upward  by  the  pressures  from  surrounding  regions.  The  inequali- 
ties of  the  earth's  surface  may  be  taken  as  approximating  to  har- 
monic undulations  of  simple  or  complex  nature.  By  so  taking  them, 
the  stresses  which  they  produce  on  the  earth's  interior  may  be 
evaluated. 

G.  H.  Darwin  treated  this  problem  in  his  paper  "On  the  Stresses 
Caused  in  the  Interior  of  the  Earth  by  the  Weight  of  Continents 
and  Mountains."1  In  this  are  investigated  the  stresses  given 
by  positive  and  negative  loads  whose  distribution  follows  a  law 

1  Phil.  Trans.  Royal  Soc.,  CLXXIII  (1882),  187-230. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  733 

of  zonal  harmonics  arranged  on  the  surface  of  a  sphere.  The 
harmonic  series  of  the  second  order  corresponds  to  oblateness  of  a 
spheroid  and  also  serves  as  a  basis  for  computing  the  tidal  strains. 
The  zonal  harmonics  of  the  fourth  order  correspond  to  an  equatorial 
continent  and  two  polar  continents,  the  eighth  order  adds  to  these 
two  annular  continents  in  about  latitude  45°,  and  so  on.  The 
higher  orders,  above  thirty,  correspond  to  a  succession  of  anti- 
clinoria  and  synclinoria,  or  mountains  and  valleys.  For  all  above 
the  second  harmonic  the  depth  of  maximum  stress  lies  within  the 
outer  half  of  the  earth's  radius. 

Darwin's  solutions  were  made  on  the  assumption  that  there  are 
no  differences  of  density  beneath  continents  and  oceans  and  that 
all  the  relief  of  the  earth  is  upheld  by  its  rigidity.  He  reached 
the  conclusion  that  continents  such  as  Africa  and  America  gave 
a  maximum  stress-difference  of  about  four  tons  per  square  inch 
at  a  depth  of  about  1,100  miles.  The  later  demonstration  of  the 
existence  of  regional  isostasy  nullifies  this  conclusion  except  for  the 
amount  by  which  the  topography  of  large  areas  is  not  completely 
compensated.  Even  this  part  can  to  some  extent  be  regarded  as 
sustained  by  a  rigid  crust  floating  upon  a  deeper  zone  which  acts 
dynamically  nearly  as  a  fluid.  There  are  reasons  for  believing, 
however,  that  this  latter  conception  is  extreme  in  the  other  direction 
and  not  justified  by  the  evidence.  It  is  thought  that  by  the  col- 
lective support  of  the  arguments  brought  out  in  this  part  the 
assumption  will  be  finally  justified — that  for  the  outstanding  loads 
not  in  isostatic  equilibrium  the  work  of  Darwin  continues  to  apply. 

In  the  mathematical  analysis,  the  loads  which  represent  the 
areas  and  heights  of  the  regional  departures  from  isostasy  are 
regarded  as  members  of  an  infinite  harmonic  series  of  ridges  and 
furrows  disposed  on  a  plane.  One  wave-length  is  the  distance 
from  crest  to  crest,  or  mid-furrow  to  mid-furrow.  Darwin  showed, 
as  illustrated  in  Figs.  15  and  16,  that  the  magnitude  of  the  stress- 
difference  at  any  point  within  the  crust  due  to  a  surficial  harmonic 
load  depended  upon  the  depth  below  the  mean  horizontal  surface 
measured  in  terms  of  the  wave-length  and  not  at  all  on  the  position 
of  the  point  considered  with  reference  to  the  ridges  and  furrows. 
Further  in  regard  to  the  direction  of  the  stress-difference,  it  is 


734  JOSEPH  BARRELL 

shown,  as  illustrated  in  Fig.  15,  that  in  passing  uniformly  and 
horizontally  through  the  crust  on  a  line  at  right  angles  to  the 
direction  of  the  ridges,  the  stress  axes  revolve  with  a  uniform 
angular  velocity.  In  relation  to  depth,  the  maximum  stress- 
difference,  as  shown  in  Figs.  15  and  16,  occurs  at  a  depth  equal  to 

-  of  the  wave-length  and  is  then  equal  to  2  gwhe'1  or  in  gravitation 

units  of  force  to  0.736^,  in  which  h  is  the  height  from  the  mean 
plane  to  the  top  or  bottom  of  the  undulations  and  w  is  the  weight 
of  a  unit  volume. 


-   X  +  X 

£         *         *         X 

I—  Wave-length 

t  TilSurface  of  maximum  stress 

32  kilometers 
$  —  Dppth  of  122  Kilometer  s 

200  kilomerers  —  J 

FIG.  15. — Diagram  showing  in  vertical  section  uncompensated  harmonic  moun- 
tains and  valleys  with  relative  magnitude  and  direction  of  stress-differences,  which 
they  impose  on  the  crust  below.  Mountain  crests  drawn  as  5  km.  above  valley 
bottoms.  Wave-length  200  km.  Stresses  shown  to  a  depth  of  122  km.  Maximum 
stress  for  this  wave-length  is  at  32  km. 

It  is  important  to  note  that  the  value  of  this  maximum  depends 
only  on  the  height  and  density  of  the  mountains  and  is  independent 
of  the  distance  from  crest  to  crest.  The  depth  at  which  this  maxi- 
mum is  reached  depends,  on  the  other  hand,  upon  the  wave-length 
and  not  upon  the  height  or  density  of  the  mountains.  The  effect 
of  a  doubling  of  the  wave-length  upon  the  vertical  distribution 
of  the  stress-differences  is  shown  in  Fig.  16. 

It  is  seen  that  the  lateral  pressure  due  to  the  elevations,  instead 
of  being  at  the  surface  as  it  would  be  under  hydrostatic  conditions 
or  as  in  completely  compensated  mountains  and  valleys  with  the 
special  distribution  of  density  assumed  by  Love,1  is  at  a  depth  of 
about  one-sixth  of  the  wave-length.  The  maximum  stress  is,  fur- 
thermore, but  37  per  cent  of  the  full  amount  of  the  hydrostatic 

1  Some  Problems  of  Geodynamics  (1911),  chaps,  ii  and  iii. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST 


735 


lateral  pressure.  Fig.  16  shows  that  uncompensated  features  with 
a  wave-length  up  to  200  km.  impose  the  stresses  almost  wholly 
within  the  lithosphere,  taking  this  as  limited  by  a  depth  of  122  km. 
The  crust  increases  in  strength  to  a  certain  maximum,  perhaps 
from  10  to  30  km.  deep,  as  shown  by  the  experiments  of  Adams. 


Surface  of  ihe  earth 


B 


lookm.-1 
Mean/depth  or  compensation 


for  uniform  disrriDurion. 


200  km.— 


o    t  Scale  for  stresses  .735 wh. 

FIG.  16. — Diagram  showing  the  distribution  in  the  crust  of  stress-differences 
due  to  parallel  uncompensated  harmonic  mountains  and  valleys:  A,  curve  showing 
relative  magnitude  and  depth  of  stress-differences  corresponding  to  a  distance  of  400 
km.  between  crests  of  parallel  mountain  ranges.  B,  curve  showing  the  same  for  a 
distance  of  200  km.  between  crests  of  parallel  mountain  ranges. 

At  greater  depth  the  strength,  according  to  the  theory  developed  in 
Part  VI,  and  as  recognized  by  Love  and  others  in  treating  of  isos- 
tasy,  is  to  be  regarded  as  decreasing  and  passing  by  transition  into 
the  asthenosphere.  Consequently  it  is  seen  that  the  distribution 
of  stress  imposed  by  a  wave-length  of  200  km.  conforms  well  to 
the  distribution  of  strength,  the  greatest  strain  coming  on  the 
strongest  part. 


736  JOSEPH  BARRELL 

Darwin  obtained  his  results  by  making  the  initial  assumption 
that  the  earth  substance  was  incompressible  but  possessed  elasticity 
of  form-  The  introduction  of  the  factor  of  compressibility  Darwin 
showed  to  affect  the  result  largely  in  the  case  of  the  second  harmonic, 
but  for  harmonics  of  infinitely  high  orders  the  resulting  stresses  are 
independent  of  the  modulus  of  compression.  Consequently  he 
states,  "it  may  be  concluded  that  except  for  the  lower  harmonic 
inequalities  compressibility  introduces  but  little  change  in  our 
results."1 

Modifications  imposed  by  long  and  large  wave-lengths. — In  Fig.  16, 
curve  A  shows  that  for  a  wave-length  of  400  km.  the  depth  of  maxi- 
mum stress  is  64  km.  and  at  122  km.  the  stress  is  75  per  cent  of  the 
maximum.  If  this  should  be  regarded  as  the  beginning  of  the 
asthenosphere  it  would  mean  that  a  large  part  of  the  stress  would 
be  thrown  on  to  the  zone  incapable  of  sustaining  large  stress- 
differences.  If  the  wave-length  became  2,000  km.  the  lithosphere 
would  be  subjected  to  but  small  stress-differences  and  the  maxi- 
mum strain  would  occur  at  a  depth  of  320  km.,  the  middle  of  the 
sphere  of  weakness.  In  order  that  Darwin's  solution  should  hold 
for  these  cases  the  height  of  the  arches  would  have  to  be  so  small 
that  the  resulting  stress-differences  would  not  exceed  the  elastic 
limit  of  the  asthenosphere.  For  greater  loads  disposed  on  the 
surface  in  these  large  wave-lengths  the  stress  relations  would 
approach  those  of  a  rigid  crust  overlying  a  fluid  substratum. 
Of  this  problem  Darwin  states,  "The  evaluation  of  stresses  in  a 
crust,  with  fluid  beneath,  would  be  tedious,  but  not  more  difficult 
than  the  present  investigation"  (on  the  stresses  caused  by  the 
weight  of  continents  and  mountains).2  It  is  a  different  problem 
from  that  solved  by  Love;  the  latter  considering  the  stresses  in 
such  a  crust  caused  by  a  condition  of  isostasy,  not  by  a  lack  of 
isostasy.  The  limited  mathematical  training  of  the  present  writer 
.  does  not  permit  here  the  definite  solution  of  this  problem,  but  some 
general  observations  can  be  made. 

For  wave-lengths  very  large  in  comparison  with  the  depth  of  the 
lithosphere  the  stress-differences,  if  confined  within  the  crust, 
approach  those  existing  in  a  continuous  beam,  each  span  being 

1  Scientific  Papers,  II,  500.  2  Ibid.,  II,  502,  footnote. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST       737 

stressed  by  a  continuous  load,  greatest  at  the  center  in  accordance 
with  the  harmonic  curve  and  acting  in  the  reverse  direction  from 
the  adjacent  spans.  This  is  a  very  simple  limiting  case.  The  maxi- 
mum bending  moment  would  be  on  the  cross-section  at  the  crest 
of  each  downward  and  upward  arch.  For  a  given  height  of  load 
the  bending  moment  would  increase  with  the  square  of  the  span 
or  wave-length.  The  maximum  bending  stresses  would  be  hori- 
zontal, acting  as  tensile  and  compressive  stresses  at  the  top  and 
bottom  of  the  lithosphere.  In  the  middle  of  the  lithosphere  there 
would  exist  a  neutral  surface  suffering  neither  tension  nor  com- 
pression, but  subjected  to  horizontal  shear.  The  theory  of  beams 
shows  that  the  strength  is  limited  by  the  marginal  tensions  and 
compressions,  not  by  the  internal  shear.  As  the  lithosphere  is, 
however,  weakest  on  the  upper  and  lower  margins,  its  material  is 
poorly  arranged  to  resist  the  bending  stresses.  The  greatest 
resistance  to  bending  in  a  certain  plane  is  given  by  the  form  of  an 
I-beam,  but  the  crust  is  analogous  to  a  beam  in  which  a  single  flange 
should  intersect  its  middle,  giving  a  cross-shaped  section.  The 
earth's  crust  is  consequently  a  peculiarly  weak  structure  to  resist 
harmonic  loads  of  great  wave-length,  and  as  the  strength  varies 
inversely  with  the  bending  moment  it  varies  inversely  with  the 
square  of  the  wave-length.  It  is  seen  then  that  wave-lengths  of 
continental  breadth  are  very  poorly  supported  by  the  strength  of 
the  crust,  but  if  they  reach  notable  amplitude  must  rest  chiefly 
upon  the  asthenosphere.  The  consideration  of  the  stress  diagram 
given  by  a  wave-length  of  200  km.  showed  why  very  pronounced 
departures  from  isostasy  can  occur  in  one  direction  over  areas  up 
to  at  least  100  km.  across  and  why  marked  regional  compensation 
extends  commonly  to  limits  of  100-200  km.  radius.  The  stress 
effects  produced  by  harmonic  loads  a  thousand  kilometers  or  more 
in  a  wave-length  show,  on  the  other  hand,  why  regional  compensa- 
tion of  the  same  vertical  magnitude  cannot  extend  effectively  across 
a  whole  continent.  It  explains  why  the  United  States  as  a  whole 
is  in  nearly  perfect  isostatic  equilibrium  with  respect  to  the  ocean 
basins. 

Nature  of  stresses  imposed  by  internal  loads. — Take  the  case  of 
harmonic  loads  distributed  on  a  plane  S-Sj  Fig.  17,  within  an 


738 


JOSEPH  BARRELL 


indefinitely  extended  elastic  solid.  The  amount  and  direction  of 
the  vertical  stress  upon  this  plane  are  shown  by  the  vertical  lines, 
the  scale  of  stresses  being  one-twentieth  of  the  value  for  the  stress- 
diagram  shown  by  the  horizontal  lines.  On  this  plane,  5-5,  a 
small  unit  mass  is  subjected  to  stress  equal  in  all  directions  and 
not  to  stress-difference,  since  the  stress  is  essentially  the  same  on 
the  small  contiguous  unit  masses.  The  reasoning  is  the  same  as 


FIG.  17. — Stress-diagrams  for  harmonic  loads  distributed  on  a  plane:  A,  diagram 
for  loads  upon  a  limiting  surface  of  a  solid  extending  indefinitely  from  this  surface. 
B,  diagram  for  loads  upon  a  plane  within  an  indefinitely  extended  solid.  The  scale 
for  loads  is  one-twentieth  of  the  scale  for  the  resulting  stresses.  A  little  more  than 
one-half  wave-length  is  shown. 

that  for  harmonic  loads  distributed  on  the  limiting  surface  of  a  solid 
except  that  here  there  is  an  indefinitely  extended  solid  on  each  side 
of  the  plane.  The  stress  at  any  point  of  the  plane  acts  positively 
on  one  side,  negatively  on  the  other.  Half  of  the  load  will  be 
carried  on  each  side.  Consequently  if  OA  is  the  curve  showing 
the  stress-differences  at  various  depths  for  a  harmonic  load  on  the 
surface  of  the  earth,  then  BOB  will  be  the  stress-curve  for  the  same 
load  carried  on  a  plane  deep  within  the  lithosphere.  For  this  to 
be  approximately  true,  however,  the  wave-length  would  have  to  be 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  739 

small  in  comparison  with  the  depth  of  the  loaded  surface.  Then 
the  upper  half  of  the  stress-diagram  will  lie  within  the  lithosphere. 
The  lower  half  of  the  curve  would  also  have  to  lie  within  the  elastic 
competence  of  the  crust  for  corresponding  depths.  Suppose  the 
plane  to  lie  near  to  either  boundary  of  the  lithosphere.  The  case 
now  approaches  that  of  a  surface  load,  one  side  of  the  stress- 
diagram  becomes  largely  cut  off  and  the  other  increases.  The 
exact  analysis  is  of  course  difficult  and  will  not  be  attempted. 

Assume  the  loaded  plane  to  be  at  a  depth  of  61  km.,  half  the 
depth  of  the  lithosphere.  The  strength  of  the  middle  would  not 
then  be  utilized  for  support  of  the  load.  For  a  wave-length  up  to 
100  km  Fig.  17  shows  fairly  well  the  distribution  of  stress  and  it 
would  be  contained  mostly  within  the  middle  of  the  lithosphere. 
For  wave-lengths  of  200  km.,  however,  the  margins  of  the  litho- 
sphere would  be  subjected  to  greater  strain  than  the  interior,  the 
stress-diagram  would  be  modified  toward  that  existing  in  a  loaded 
beam,  and,  if  the  margins  are  weak,  the  structure  is  poorly  adapted 
to  support  the  load.  If  the  loaded  plane  is  at  greater  depth,  the 
same  wave-length  will  throw  a  greater  proportion  of  the  strain 
upon  the  asthenosphere  and  the  deeper  parts  of  the  lithosphere. 
If  these  are  incapable  of  supporting  the  resulting  stresses,  again 
a  modification  of  the  diagram  would  occur,  involving  bending 
moments  in  the  stronger  part  of  the  crust. 

If,  now,  it  be  assumed  that  the  upper  half  of  the  lithosphere  is 
decidedly  stronger  than  the  lower  half  and  that  the  maximum 
strength  is  at  some  depth  below  the  surface,  it  is  seen  that  the 
maximum  outstanding  masses  which  the  crust  could  carry  would 
be  disposed  in  the  outer  quarter  or  third  of  the  lithosphere.  But 
this  is  just  what  was  found  to  be  the  case  as  the  result  of  the  studies 
made  in  Part  V.  Therefore  the  accordance  between  the  geodetic 
evidence  and  the  consequences  of  the  assumption  raise  it  to  the 
dignity  of  a  presumption.  It  may  be  taken  as  a  working  hypothe- 
sis that  the  greater  outstanding  masses  are  limited  in  their  positions 
by  the  limitations  of  crustal  strength.  Mental  reservation  must 
be  made,  however,  as  to  the  possibility  of  other  more  important 
determining  factors,  such  for  example  as  the  nature  of  igneous 
activity,  in  limiting  the  zone  of  large  outstanding  masses.  An 


740  JOSEPH  BARRELL 

accordance  of  fact  with  theory  is  not  a  proof,  but  it  raises  a  pre- 
sumption that  the  theory  is  correct. 

Nature  of  stresses  imposed  by  perfect  isostasy. — This  topic, 
although  not  directly  in  line  with  die 'subject  of  this  chapter, 
must  receive  brief  mention  here  since  the  stresses  in  the  crust  are 
compounded  of  those  due  to  the  departures  from  isostasy  with 
those  resulting  from  a  state  of  perfect  isostasy.  The  stress  resulting 
from  the  isostatic  support  of  continents  and  mountains  has  been 
ably  worked  out  by  Love.1  But  his  treatment  started  with  the 
limiting  though  improbable  assumption  that  at  a  depth  of  one- 
fiftieth  of  the  earth's  radius  all  stress-differences  disappeared,  as 
though  the  layer  below  were  truly  fluid.  This  required  a  com- 
plicated and  equally  improbable  curve  of  density,  opposite  in 
sign  above  and  below,  in  order  that  the  topography  should  be  com- 
pensated, the  ocean  surface  remain  a  level  surface,  and  yet  the 
stress-differences  become  zero  at  the  required  depth.  Neverthe- 
less the  solution  is  valuable  as  a  limiting  case  in  showing  the  general 
character  of  the  internal  stresses  which  must  exist.  He  showed 
that  isostatically  compensated  harmonic  mountains  and  valleys 
gave  maximum  stress-differences  on  the  axial  lines  of  mountains 
and  valleys  and  that  it  amounted  to  about  one-fourth  of  the 
theoretical  hydrostatic  pressure.  The  stresses  decrease  rapidly 
with  depth. 

In  contrast  to  Love's  hypothesis  of  the  distribution  of  density, 
that  of  Hay  ford  may  be  considered.  This  is  that  the  excess  or 
defect  in  density  needed  for  compensation  is  uniformly  distributed 
to  a  depth  of  122  km.  Again,  a  rigorous  mathematical  treatment 
must  be  left  to  those  competent  to  undertake  it,  but  it  would  appear 
that  such  distributions  of  density  would  throw  very  considerable 
stress-differences  within  the  asthenosphere ;  or,  if  this  was  incompe- 
tent to  carry  such,  would  bring  large  stress-differences  upon  the 
bottom  of  the  zone  of  compensation,  opposite  in  sign  to  that  in 
the  upper  half,  whereas  at  an  intermediate  level  depending  upon 
the  wave-length  would  be  a  region  of  no  stress-difference.  This 
distribution  of  stress  resulting  from  the  hypothesis  may  be  taken 
as  a  strong  argument  against  the  existence  of  a  uniform  distribution 

1  Some  Problems  of  Geodynamics  (1911). 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  741 

to  the  isostatic  compensation.  In  order  to  have  the  larger  relief 
and  its  compensation  fitted  to  a  crust  strongest  in  its  upper  part 
and  shading  into  a  zone  of  weakness,  the  zone  of  compensation 
should  die  out  with  deptb  ai*ji  lie  mostly  in  the  upper  half  of  the 
zone  of  strength,  since  the  stress-differences  would  die  out  at  a 
depth  greater  than  the  disappearance  of  the  compensation.  This 
conclusion  is  seen  to  be  in  closer  accord  with  several  other  lines  of 
evidence  which  have  been  noted  than  is  the  contrary  assumption 
of  uniform  compensation. 

[To  be  continued] 


THE  STRENGTH  OF  THE  EARTH'S  CRUST 


JOSEPH  BARRELL 
New  Haven,  Connecticut 


PART  VII.    VARIATIONS  OF  STRENGTH  WITH  DEPTH  AS  SHOWN 
BY  THE  NATURE  OF  DEPARTURES  FROM  ISOSTASY     . 

SECTION  B,  APPLICATIONS  OF  THE  THEORY 

GEODETIC  EVIDENCE  AS  TO  LIMITING  HEIGHTS  AND  WAVE-LENGTHS  .  27 

Measurements  of  Strength  by  Maximum  Loads 27 

Harmonic  Loads  with  Short  Wave-Lengths 28 

Maximum  Loads  for  Mean  Wave-Lengths 29 

Departures  from  Isostasy  of  Large  Wave-Lengths 30 

RELATIONS  or  ACTUAL  STRESSES  TO  THE  SUM  OF  HARMONIC  WAVES  32 

GEOLOGIC  SUGGESTIONS  AS  TO  MAGNITUDES  OF  CRUSTAL  STRESSES  35 

Submarine  Geanticlines  and  Geosynclines   ..........  35 

The  Niger  Delta 38 

The  Existing  Continental  Ice  Sheets 38 

Accordance  of  Geologic  with  Geodetic  Evidence 40 

ADJUSTMENT  OF  LOADS  TO  THE  DISTRIBUTION  OF  STRENGTH     ...  40 

CHARACTER  OF  THE  CURVE  OF  STRENGTH 42 

GEODETIC   EVIDENCE   AS    TO   LIMITING   HEIGHTS   AND 
WAVE-LENGTHS 

Measurements  of  strength  by  maximum  loads. — In  the  course  of 
geologic  time  the  internal  forces  of  igneous  intrusion  and  tangential 
compression,  the  external  forces  of  erosion  and  sedimentation,  have 
tended  to  strain  the  crust  to  its  limits  of  strength,  and  the  degree 
of  isostasy  which  exists  constitutes  a  measure  of  those  limits. 
Small  loads  of  large  wave-length  and  large  loads  of  small  wave- 
length will  tend  to  rise  to  maxima  which  may  be  used  in  connection 
with  the  theory  of  distribution  of  stress,  as  considered  in  Part  VII, 
Section  A,  to  give  an  approximate  idea  of  the  limits  and  distribution 
of  strength  in  the  crust. 

The  problem  is  to  find  the  maximum  vertical  load  and  its  rela- 
tion to  wave-length  acting  over  areas  which  may  be  regarded  as 

27 


28  JOSEPH  BARRELL 

forming  roughly  a  harmonic  series.  The  theory  applies  best  where 
elongated  unit  areas  are  flanked  by  areas  of  opposite  sign.  But 
even  where  a  single  axis  of  uncompensated  elevation  or  depression 
is  surrounded  with  a  region  of  mean  elevation  it  may  be  regarded 
as  a  half  of  a  wave  sustained  by  the  rigidity  of  the  earth.  The 
stresses  in  the  vertical  plane  through  its  crest  line  would  appear 
to  be  less  than,  but  not  so  greatly  different  from,  what  they  would 
be  for  a  continued  series.  Where  the  load  is  of  oval  instead  of 
zonal  distribution  the  stresses  would  also  be  somewhat  diminished 
in  case  the  oval  is  surrounded  by  a  neutral  region,  but  if  a  series 
of  ovals  of  opposite  signs  is  analogous  to  two  intersecting  wave- 
series,  although  the  stresses  would  be  complicated  and  are  not  in 
general  the  sum  of  the  stresses  due  to  the  separate  series,  yet  it 
does  not  appear  that  the  extreme  maxima  would  be  necessarily 
less  than  the  sum,  and  many  of  the  maxima  would  be  greater  than 
the  maxima  of  the  separate  series. 

Harmonic  loads  with  short  wave-lengths. — Horizontal  compres- 
sion builds  mountain  folds  of  which  the  individual  ranges  are 
clearly  the  results  of  compression  and  not  of  isostatic  elevation. 
Erosion  dissects  an  elevated  country  on  a  pattern  of  a  certain  scale, 
deep  valleys  of  erosion  separating  the  crust  remnants  above  base- 
level  which  are  not  yet  consumed.  These  actions  produce  a  surface 
relief  which  corresponds  roughly  to  various  harmonic  series  of 
appreciable  amplitude  and  wave-length,  but  in  this  section  will  be 
considered  only  the  geodetic  evidence  of  variable  mass  not  iso- 
statically  compensated,  much  of  the  variation  being  due  to  the 
concealed  heterogeneities  of  density. 

The  distance  between  Washington,  B.C.,  and  Hoboken,  New 
Jersey,  as  estimated  in  Part  V,  Section  B,  is  326  km.  Within  this 
distance  are  three  intermediate  stations  and  the  two  limiting 
stations,  each  station  showing  a  gravity  anomaly  opposite  in  sign 
to  that  of  the  adjacent  stations.  There  must  be  then  at  least 
two  wave-lengths.  The  average  change  of  anomaly  between  the 
adjacent  stations  is  0.021  dyne.  As  it  is  wholly  improbable  that 
the  stations  are  located  at  the  crest  lines  of  the  waves,  the  whole 
amplitude  may  safely  be  taken  as  at  least  o .  026  and  the  wave- 
length 1 60  km. 


•     THE  STRENGTH  OF  THE  EARTH'S  CRUST  29 

Seattle  and  Olympia  are  80  km.  apart  and  the  difference  of 
their  anomalies  is  o .  1 26  dyne.  If  the  anomalies  had  been  measured 
at  the  points  giving  maximum  values  they  would  certainly  give  a 
difference  of  at  least  o.  130  or  perhaps  o.  140,  about  five,  times  the 
amplitude  of  the  variations  between  Washington  and  Hoboken. 
This  large  value  seems,  however,  to  be  exceptional  for  the  United 
States  and  may  constitute  but  a  single  wave.  We  may  take  it, 
however,  as  showing  that  the  crust  can  sustain  a  harmonic  load  of 
1 60  km.  (100  miles)  wave-length  and  total  amplitude  of  0.120 
dyne.  For  the  reasons  explained  in  Part  IV,  especially  on  p.  304, 
the  divisor  to  be  used  to  turn  this  anomaly  into  the  equivalence 
of  rock  measured  in  feet  could  not  be  over  0.0018  and  a  better 
figure  for  the  interpretation  of  this  short  wave-length  is  0.0015. 
This  gives  an  amplitude  of  8,000  feet  (2,440  m.).  The  part  below 
the  mean  level  is  then  50  miles  wide  and  4,000  feet  deep,  the 
adjacent  positive  parts  of  the  wave  being  of  equal  dimensions  but 
in  the  opposite  direction.  This  is  of  the  same  order  of  relief  as 
the  larger  ranges  of  folded  mountains  and  intermontane  valleys. 
The  stresses  which  this  harmonic  series  imposes  upon  the  crust  are 
shown  by  curve  A  of  Fig.  18. 

Maximum  loads  for  mean  wave-lengths. — In  Part  II  it  was  argued 
that  the  evidence  of  anomalies  from  mountain  stations  showed 
regional  compensation  on  the  average  probably  to  the  outer  limit 
of  zone  O,  radius  166.7  km.,  diameter  consequently  333.4km. 

Over  the  United  States  in  general  the  intercepts  of  the  areas 
of  grouped  deflections  averaged  180  miles.  The  average  diameter 
would  therefore  doubtless  be  as  great  as  200  miles  (3 20 km.). 

On  Bowie's  map  of  the  New  Method  gravity  anomalies,1  it  is 
seen  that  the  distances  from  pronounced  maxima  to  pronounced 
minima  average  250-350  km. 

From  these  several  lines  of  evidence  we  may  conclude  with  some 
confidence  that  the  half  wave-length  for  pronounced  anomalies  in 
the  United  States  averages  near  300-3  50  km.  The  wave-length 
is  therefore  600-700  km.  (373-435  miles).  A  wave-length  of 
600  km.  will  be  taken.  The  pronounced  maxima  for  these  waves 
runs  from  plus  or  minus  0.030  to  0.060  dyne.  The  real  maxima 

1  This  article,  Part  II,  Jour.  GeoL,  XXII  (1914),  153. 


30  JOSEPH  BARRELL 

are  to  be  regarded  in  most  cases  as  situated  to  one  side  of  the  stations 
and  somewhat  greater.  But  exceptional  and  local  maxima  must 
be  smoothed  out  to  form  a  part  of  the  harmonic  curve.  It  is, 
furthermore,  the  difference  of  anomaly  between  adjacent  opposite 
phases  which  is  the  significant  feature.  This  difference  runs  from 
0.060  to  0.080.  The  latter  figure  will  be  chosen.  For  this  wave- 
length, representing  a  certain  unit  area  of  attraction,  the  best 
divisor  is  perhaps  to  take  0.0024  dyne  of  anomaly  as  equivalent  to 
100  feet  of  rock.  An  anomaly  of  o .  080  dyne  is  on  that  basis  equiva- 
lent to  3,330  feet  (1,015  m.)  °f  rock-  The  crust  of  the  United 
States  sustains,  therefore,  harmonic  loads  600  km.  (373  miles)  in 
wave-length  and  1,015  m-  (3>33°  ^eet)  m  total  amplitude.  The 
stresses  which  this  wave-series  imposes  on  the  crust  are  shown  by 
curve  B,  Fig.  18. 

Departures  from  isostasy  of  large  wave-lengths. — For  the  continent 
as  a  whole  and  in  its  relations  to  the  ocean  basins  isostasy  is  nearly 
perfect;  but  the  question  rises  here,  how  nearly?  The  first  term 
of  the  gravity  formula  for  the  Vienna  system  of  gravity  observations 
is  978.046  dynes.  The  first  term  for  the  Potsdam  system  is 
978.030.  The  first  term  for  the  United  States  system  after  reject- 
ing the  Seattle  anomalies  is,  as  shown  by  Bowie,  978.038  dynes. 
These  respective  systems  differ  as  a  whole  by  these  amounts.  We 
have  no  right  to  assume  that  any  one  is  absolutely  correct.  The 
whole  of  the  United  States  system  may  lie  a  little  above  or  below 
the  level  giving  isostatic  compensation  with  respect  to  the  average 
surrounding  ocean  basins,  or  with  respect  to  the  entire  earth.  The 
mean  value  for  the  United  States  suggests,  however,  that,  as  a 
whole,  the  continent  lies  within  a  few  hundred  feet,  possibly  less 
than  one  hundred  feet,  of  the  level  which  would  give  perfect 
isostatic  equilibrium. 

Let  us  consider  next  its  larger  parts.  These  can  be  compared 
with  each  other  and  with  the  United  States  as  a  whole.  Although, 
as  discussed  in  Part  IV,  the  map  of  gravity  anomalies  lacks  detail, 
the  grouping  of  many  stations  of  like  sign  into  large  areas  gives 
confidence  in  the  conclusion  that  there  are  regional  departures 
from  isostasy.  These  are  of  two  or  three  orders  of  magnitude,  of 
which  the  areally  smaller  have  been  discussed.  To  bring  out  the 


THE  STRENGTH  OF  THE  EARTH'S  CRUST        31 

areally  larger  we  must  draw  boundaries  about  large  regions  which 
show  a  dominance  of  anomalies  of  one  sign.  These  boundaries, 
however,  must  be  taken  so  as  to  give  compact  unit  areas,  so  as  not 
to  obtain  an  unreal  result  by  the  political  expedient  of  gerrymander- 
ing the  districts. 

Select  as  a  center  the  point  whose  geographic  co-ordinates  are 
lat.  42°,  long.  102°.  Describe  about  this  center  a  circle  of  850  km. 
radius.  This  includes  an  area  equal  to  29  per  cent  of  the  area  of 
the  United  States.  It  should  be  taken  as  including  the  negative 
anomaly  station  99  on  its  southern  border.  This  circle  covers  a 
large  positive  region  which  could  be  made  still  more  positive  by  an 
extension  of  its  boundaries  to  the  northeast  over  Wisconsin  and 
Michigan.  Within  this  circle  are  distributed  with  a  fair  degree  of 
uniformity  31  of  the  122  gravity  stations  of  the  United  States. 
The  mean  with  regard  to  sign  of  the  anomalies  of  these  31  stations 
referred  to  the  United  States  mean  with  regard  to  sign  is  +  o.oio 
dyne.  As  the  mean  without  regard  to  sign  of  all  stations  in  the 
United  States  excluding  Seattle  is  only  0.018,  it  is  seen  that  this 
positive  region  stands  out  clearly  from  the  general  average. 

West  of  this  circle  and,  on  the  south,  to  the  west  of  long.  107° 
there  are  21  stations,  including  one  of  the  two  Seattle  stations. 
These  mark  a  broad  region  of  negative  anomaly.  The  mean 
anomaly  with  regard  to  sign  is  —0.017  dyne.  There  seems  to  be 
no  reason  for  completely  omitting  the  exceptionally  large  Seattle 
anomalies.  One  of  them  has  therefore  been  retained,  but  if  both 
are  omitted  the  mean  is  still  —0.013.  The  value  of  —0.017 
will  here  be  adopted.  The  difference  of  the  means  of  the  central 
and  western  regions  is  consequently  0.027  dyne.  Let  these  be 
regarded  as  the  positive  and  negative  phases  of  an  harmonic  wave 
and  the  mean  departure  of  the  two  phases  becomes  0.0135  from 
each  side  of  the  mean  plane.  Now  it  may  be  computed  for  a 
harmonic  wave  represented  by  the  formula  y—A  sin  Bx  that  the 
mean  height  of  the  wave  above  the  mid-plane  is  64  per  cent  of  the 
crest  height.  From  mid-plane  to  crest  of  this  wave- series  is  there- 
fore 0.021.  From  the  large  negative  anomalies  along  the  Pacific 
coast  it  would  seem  that  this  negative  zone  must  extend  somewhat 
further.  The  wave-length  of  this  series  is  consequently  between 


32  JOSEPH  BARRELL 

2,600  and  3,000  km.  A  mean  value  of  2,800  km.  (1,740  miles)  will 
be  chosen.  From  the  breadth  of  half  a  wave-length  it  appears  that 
0.0034  dyne  of  anomaly  may  be  taken  as  equivalent  to  100  feet  of 
rock.  This  gives  the  crest  and  trough  as  625  feet  (190  m.)  from 
the  mean  plane,  a  total  amplitude  of  1,250  feet  (380  m.).  The 
stress-differences  which  this  wave-series  throws  upon  an  earth 
elastically  competent  throughout  to  bear  the  stresses  are  shown 
by  curve  C,  Fig.  18.  Helmert  has  published  an  extensive  paper 
dealing  with  the  force  of  gravity  and  the  distribution  of  mass  in  the 
crust  of  the  earth,1  to  which  the  writer's  attention  has  been  called 
recently  by  Professor  Pierpont,  of  the  mathematical  department 
of  Yale  University.  In  this  paper  Helmert  adopts  the  hypothesis  of 
regional  isostasy  and  finds  his  results  confirmatory  of  it,  but  not 
in  accord  with  the  hypothesis  of  close  and  local  isostatic  adjustment. 
His  work  is  especially  valuable  as  confirmatory  of  the  present  con- 
clusions, since  it  deals  with  regions  outside  of  the  United  States.  As 
he  does  not,  however,  compute  the  corrections  due  to  the  distant 
large  irregularities  of  topography,  his  figures  cannot  be  directly 
compared  with  Hayford's  New  Method  anomalies.  Neverthe- 
less his  conclusions  as  to  the  existence  of  broad  regional  excesses 
or  defects  of  mass  are  comparable  to  those  here  reached.  Under 
the  section  on  the  horizontal  displacement  of  compensation  and 
extended  excesses  and  defects  of  mass2  he  sums  up  part  of  the 
evidence  in  the  following  statement:  "We  have  then  to  deal  with  a 
continuous  region  of  positive  total  gravity  disturbance  in  Europe 
1,000  km.  broad  and  also  with  a  region  of  negative  disturbance 
in  Asia  of  at  least  500  km.  breadth,  both  possessing  great  linear 
extension." 

RELATIONS  OF  ACTUAL  STRESSES  TO  THE  SUM  OF  HARMONIC  WAVES 

Both  Darwin  and  Love  point  out  that  the  actual  stress- 
differences  imposed  by  the  superposition  of  different  harmonic 
waves  is  not  in  general  the  sum  of  the  individual  stress-differences. 
Darwin,  however,  states  the  special  conditions  under  which  the 

1  "Die  Schwerkraft  und  die  Massenverteilung  der  Erde,"  Ency.  Math.  Wissen- 
schaft,  Band  VI,  i,  B,  Heft  2  (1910),  pp.  85-177. 

2  Op.  cit.,  pp.  152-54. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST 


33 


resultant  is  the  sum  of  the  individual  stress-differences.1  The  three 
waves  which  have  been  considered  are  types  which  coexist  and  are 
superimposed.  The  total  stress  which  they  give  would  vary  from 

7/oc  founds  per  square  inch  1,000  o 


soo 


Kilograms  per  square  cenH merer 


100 


•Surface  of  Hie  earf-h 


Lirhosphere 


Asthenosphere 


Centre  sphere 


o  km 


100. 


200. 


300. 


400. 


500, 


600. 


700. 


800 


FIG.  18. — Stress-curves  for  harmonic  waves  on  an  earth  elastically  competent 
throughout,  the  waves  representing  departures  from  isostasy  in  the  United  States 
as  given  by  analyzing  the  geodetic  data  into  the  following  harmonic  waves: 

A,  wave-length  160  km.,  amplitude  o.  120  dyne  =  2,440  m.  of  rock. 

B,  wave-length  600  km.,  amplitude  0.080  dyne  =  1,015  m.,  of  rock. 

C,  wave-length  2,800  km.,  amplitude  0.042  dyne  =  380  m.  of  rock. 
D  is  the  sum  of  A,  B,  and  C. 

E,  wave-length  400  km.,  amplitude  0.366  dyne  =  4,000  m.  of  rock  (from  the 
Pacific  Ocean). 

F',  curve  of  strength  suggested  by  geodetic  evidence  from  the  United  States. 

F",  curve  of  strength  suggested  by  geologic  evidence  from  various  parts  of  the 
world. 

complete  neutralization  up  to  their  sum  as  a  possible  maximum. 
Curve  D  represents  such  an  addition  of  A,  B,  and  C,  Fig.  18.  There 
are  reasons  why  this  curve  may  be  taken  as  a  fairer  representation 
of  the  maximum  stress  conditions  under  the  United  States  than 

1  Scientific  Papers,  II,  492. 


34  JOSEPH  BARRELL 

any  individual  curve,  even  though  there  may  be  no  place  where 
the  culminating  phases  of  like  sign  all  coincide  and  become  additive. 
These  reasons  are  found  in  the  subsurface  location  of  those  loads 
due  to  outstanding  density  and  also  to  the  added  stresses  due  to 
isostatic  compensation.  These  causes  result  in  throwing  a  greater 
stress  upon  the  outer  parts  of  the  lithosphere  and  also  serve  to 
broaden  downward  the  stress  diagrams.  Furthermore,  the  stresses 
due  to  isostatic  compensation  of  continents  would  appear  to  be  in 
reality  much  greater  under  the  margins  than  the  small  values  com- 
puted by  Love,1  since  he  has  taken  the  continents  as  having  the 
broad  sweeping  surfaces  of  a  harmonic  nature,  whereas,  as  a  matter 
of  fact,  the  continents  slope  off  rather  abruptly  to  the  depths  of  the 
oceans.  Facing  the  Pacific  in  fact,  the  two  Americas  show  high 
mountain  elevations.  This  would  cause  the  stresses  in  the  vicinity 
of  these  continental  margins  to  resemble  those  imposed  by  a  great 
mountain  chain  and  its  isostatic  compensation  rather  than  those 
imposed  by  the  breadth  of  a  continent.  If  isostatic  compensation 
is  complete  under  mountain  slopes,  Love  shows  for  the  cases  com- 
puted by  him  that  the  maximum  stress  is  about  equal  to  that  given 
by  a  column  of  rock  one-fourth  the  total  height  from  mountain 
crest  to  valley  bottom.  If  the  abyssal  slopes  of  the  continental 
platforms  be  taken  as  averaging  3-4  km.  in  elevation  and  50-100 
km.  in  width,  it  is  seen  that,  even  if  fully  compensated,  they  add 
stresses  to  the  crust  which  may  approach  in  magnitude  one-half 
of  the  stresses  shown  by  curve  A  of  Fig.  18.  The  extreme  depths 
of  slope  are  much  greater  and  it  is  clear  that  isostatic  compensation 
cannot  be  exact  under  these  great  reliefs.  Therefore  we  may  con- 
clude that  curve  D  does  not  overestimate  the  maximum  stresses 
imposed  by  the  irregularities  of  the  crust,  both  compensated  and 
uncompensated,  as  indicated  by  geodetic  evidence  within  the 
United  States  and  especially  along  its  ocean  borders.  This  investi- 
gation, however,  has  been  of  a  general  nature  and  is  designed 
merely  to  establish  an  order  of  magnitude.  It  remains  for  future 
work  to  make  more  precise  analyses  for  each  locality  from  the  data 
which  may  be  acquired,  and  especially  to  investigate  quantitatively 
the  problems  offered  by  critical  areas. 

1  Some  Problems  of  Geodynamics  (1911),  chap.  ii. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  35 

GEOLOGIC   SUGGESTIONS   AS   TO   MAGNITUDES    OF   CRUSTAL   STRESSES 

Submarine  geanticlines  and  geosyndines. — Here  will  be  con- 
sidered some  geologic  illustrations  of  departures  from  isostasy, 
arranged  in  order  of  harmonic  wave-length.  They  are  to  be  com- 
pared with  the  results  obtained  from  the  study  of  the  geodetic 
data.  Most  of  the  geologic  evidence  is  merely  suggestive,  not 
conclusive,  since  diametrically  opposite  opinions  are  held  as  to  the 
probability  of  the  visible  load  being  offset  by  an  invisible  com- 
pensation. As  suggestions,  however,  they  are  none  the  less  valuable, 
and  point  the  way  to  needed  geodetic  observations. 

The  mountain  folds  advance  from  Asia  over  the  floor  of  the 
Pacific  Ocean,  forming  the  system  named  by  Suess  the  Oceanides. 
Mostly  hidden  beneath  the  ocean  surface,  they  have  been  but  little 
affected  by  erosion.  Their  ridges  and  deeps  mark  the  greatest 
mountain  reliefs  of  the  globe.  It  is  probable  that  here,  if  anywhere, 
tangential  pressures  have  forced  the  crust  into  folds  whose  height 
combined  with  span  is  as  great  as  the  strength  of  the  crust  can 
endure.  To  what  degree  the  elevations  and  depressions  are  com- 
pensated is,  however,  unknown,  and  the  great  arches  are  supported 
in  part  by  the  lateral  pressure  of  the  ocean  water.  It  is  quite  pos- 
sible if  not  probable  that  appreciable  changes  of  deep-seated 
density  may  accompany  the  growth  of  such  ridges,  especially  as 
they  mostly  exhibit  a  volcanic  activity  and  are  to  a  greater  or  less 
extent  structures  built  up  by  igneous  extrusion.  It  is  not  at  all 
probable,  however,  that  they  are  completely  or  possibly  even 
largely  compensated,  but  where  the  mountain  folds  and  trough-like 
deeps  broaden  into  plateaus  or  anti-plateaus  the  presumption  is 
strengthened  that  the  forms  may  there  be  isostatically  compen- 
sated to  a  large  degree.  Such  plateaus  or  anti-plateaus  cannot 
then  be  used  in  the  present  argument.  The  ridges  and  troughs, 
however,  show  in  their  forms,  as  has  been  stated,  modes  of  con- 
struction which  are  not  conditioned  on  isostasy.  Let  attention 
be  turned  then  to  the  folds  of  the  ocean  floor. 

Passing  from  west  to  east,  first  may  be  noted  between  the 
Philippines,  Borneo,  and  New  Guinea  a  complex  of  ridges  and 
basin-like  .deeps.  The  larger  wave-length  of  that  region  runs  from 
300  to  500  km.  The  Ladrone  Islands  and  Nero  Deep  give  a 


36  JOSEPH  BARRELL 

distance  of  about  150  km.  from  crest  to  trough  and  a  wave-length 
of  400-500  km.,  these  folds  and  many  others  exhibiting  a  lack  of 
symmetry.  The  existence  of  strong  folding  pressures  and  a 
tendency  to  overthrusting  and  secondary  vertical  faulting  seem  to 
be  expressed  therein.  The  great  fold  passing  north  from  New 
Zealand  and  showing  as  the  Kermadec  and  Tonga  islands  with 
their  fore-deeps  gives  a  distance  from  crest  to  trough  of  1 20-180  km., 
a  wave-length  of  400-500  km.  Lower  California  and  the  troughs 
on  each  side  show  a  wave-length  of  350  km.  The  region  of  the 
Lesser  Antilles  is  tectonically  a  northward  branching  of  the  Andean 
mountain  system  and  shows,  like  the  folds  of  the  Pacific,  crustal 
undulations  with  a  wave-length  of  350  km.  We  may  conclude 
then  that  these  folds  of  the  ocean  floor  have  a  marked  tendency  to  a 
wave-length  of  300-500  km.,  there  being  commonly  one  great  asym- 
metric fold  passing  out  into  subordinate  marginal  folds.  The  vol- 
canic chain  of  the  Hawaiian  Islands  shows,  however,  no  related  deeps 
and  has  a  half  wave-length  of  about  200  km. 

Hayford  and  Bowie  have  given  the  New  Method  anomalies 
for  a  few  stations  in  these  regions.1  A  portion  of  the  data  has 
been  abstracted  and  given  in  Part  IV  of  the  present  article.2 
Four  observations  of  Hecker's  for  the  Tonga  Plateau  and  Tonga 
Deep  are  given.  They  may  not  be  of  high  value,  since  the 
method  has  been  criticized  as  not  possessing  accuracy  compar- 
able to  observations  made  by  pendulum  upon  land.  Further- 
more, the  four  observations,  two  over  the  plateau  and  two  over 
the  deep,  are  spread  through  a  distance  of  5,100  km.  along  the  axis 
of  the  structure  instead  of  being  taken  on  a  transverse  section. 
Nevertheless,  as  the  reliefs  and  the  corresponding  anomalies 
are  all  of  great  magnitude,  the  errors  become  relatively  small  and 
the  mean  of  the  observations  is  therefore  of  some  value.  The  two 
New  Method  anomalies  for  the  Tonga  Plateau  give  a  mean  of 
+0.202  dyne,  the  depth  of  water  being  2,700:01.  The  two  New 
Method  anomalies  for  the  Tonga  Deep  give  a  mean  of  —0.172 
dyne,  the  depth  of  water  averaging  7,5oom.  If  the  amplitude 

1  The  Effect  of  Topography  and  Isostatic  Compensation  upon  the  Intensity  of  Gravity, 
1912,  p.  81. 

*Jour.  GeoL,  XXII  (1914),  311. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  37 

of  the  uncompensated  portion  of  the  crust-waves  be  measured  in 
terms  of  anomaly  by  taking  the  algebraic  sum  of  the  anomalies 
over  the  plateau  and  the  deep,  a  total  amplitude  is  obtained  of 
0.374  or  a  half-amplitude  of  0.187.  On  the  island  of  Hawaii 
an  observation  on  Mauna  Kea  at  an  elevation  of  3,981  m.  gave  a 
New  Method  anomaly  of  +0.183,  almost  the  same  figure  as  the 
half-amplitude  for  the  great  Tonga  crust- wave. 

Helmert  has  discussed  the  gravity  disturbances  found  in  the 
Hawaiian  Islands  and  states  of  them:  "For  the  Hawaiian  Islands 
it  must  be  concluded  on  the  whole  that  a  part  of  the  mass  gives 
rise  to  positive  gravity  disturbances  and  only  the  remainder  is 
isostaticalry  supported.  If  the  disturbances  were  produced  solely 
by  the  mass  of  the  islands  the  values  of  Ag  and  Ag"  [the  disturb- 
ances of  gravity]  would  be  somewhat  greater  than  they  are  found."1 

From  this  review  of  the  mountain  chains  of  the  Pacific  it  may  be 
concluded  that  the  ocean  floor  can  sustain  a  harmonic  wave-length 
of  400  km.  which  gives  an  anomaly  at  the  crest  lines  as  great  as 
that  observed  on  Mauna  Kea,  0.183  dyne.  To  interpret  this  as 
an  equivalent  load  of  rock  a  divisor  must  be  selected.  The  divisor 
depends  upon  the  depth  and  distribution  of  compensation  and  the 
area  of  the  region  of  outstanding  mass.  As  shown  in  Part  IV, 
p.  311,  0.0024  dyne  might  reasonably  be  chosen  as  the  amount  of 
anomaly  equivalent  to  100  feet  of  rock,  but  for  these  great  loads  it 
is  desirable  to  lean  toward  the  side  of  an  underestimate.  Therefore 
0.0030  will  be  taken  as  the  divisor.  This  gives  1,868  m.  as  the 
crest  height  of  the  uncompensated  portion  of  the  Hawaiian  moun- 
tain chain.  The  same  applies  to  the  Tonga  fold.  If,  however, 
o. 0024  should  be  chosen  as  the  divisor,  then  o.  183  dyne  of  anomaly 
would  correspond  to  a  half-amplitude  of  2,334m. 

It  may  be  taken  then  as  fairly  certain  that  these  great  moun- 
tain chains  show  reliefs  which  depart  as  much  as  2,000  m.  above  and 
below  the  mean  level  which  would  give  perfect  isostasy.  It  may 
be  concluded  in  consequence  that  the  oceanic  crust  can  sustain 
a  harmonic  wave-length  of  400  km.  with  an  uncompensated 
amplitude  m'easured  by  4,000  m.  of  rock.  The  diagram  of 

'"Die  Schwerkraft  und  die  Massenverteilung  der  Erde,"  Ency.  math.  Wissen- 
schaften,  Band  VI,  i,  B,  Heft  2,  (1910),  p.  133. 


38  JOSEPH  BARRELL 

stress-differences  for  this  is  shown  in  curve  E,  Fig.  18.  But  if  the 
rock  has  a  density  of  2.67  and  the  sea- water  a  density  of  1.03, 
this  corresponds  to  an  amplitude  beneath  the  ocean  surface  of 
6,513  m.  of  uncompensated  rock.  This  is  only  about  two- thirds  of 
the  maximum  relief  which  is  observed,  so  that  it  is  well  within  the 
limits  of  possibility.  These  few  available  figures  suggest  that  the 
sharp  submarine  ridges  and  deeps  may  not  be  more  than  one-third 
or  two- thirds  compensated. 

The  Niger  delta. — Reverting  to  the  discussion  of  the  Niger  delta 
given  in  Part  I,  it  is  seen  that  there  is  no  evidence  of  depression 
around  its  margin.  It  may  be  taken  then  as  the  positive  half  of 
a  harmonic  wave  well  within  the  limits  of  crustal  strength.  If  the 
section  of  the  delta  be  taken  as  given  in  Figs.  3  and  4,  pp.  31,  43,  it 
is  seen  that  the  load  is  disk-like  in  form,  instead  of  being  indefinitely 
elongated  at  right  angles  to  the  section  in  accordance  with  the 
form  of  a  zonal  harmonic.  It  seems  likely,  because  of  these  two 
departures  from  the  nature  of  a  harmonic  series,  that  the  stresses 
beneath  it  are  not  more  than  half  of  those  which  would  be  given 
by  the  completed  harmonic  curve.  As  it  is  merely  the  order  of 
magnitude  of  the  stress-differences  which  we  may  hope  to  attain 
we- may  proceed  in  accordance  with  these  rough  assumptions. 
It  is  seen  that  the  section  of  the  delta  shows  a  half  wave-length 
of  about  300  km.  and  a  maximum  thickness  equivalent  to  1,650  m. 
of  rock  upon  land.  This  corresponds  to  the  half -amplitude  or 
thickness  above  the  mean  plane.  If  half  of  this  is  taken  as  a 
measure  of  the  stress,  it  gives  a  wave-length  of  600  km.  and  a 
total  amplitude  of  i,65om.  The  stress-curve  for  this  harmonic 
series  is  60  per  cent  larger  than  the  stresses  due  to  the  outstanding 
masses  of  the  same  wave-length  as  given  by  the  geodetic  evidence 
in  the  United  States  and  shown  in  curve  B,  Fig.  18.  As  the  esti- 
mate from  the  Niger  delta  is  very  imperfect  and  unchecked  by 
pendulum  observations  reduced  by  the  New  Method,  the  stress- 
curve  is  not  plotted. 

The  existing  continental  ice  sheets. — Two  ice  sheets  of  sub- 
continental proportions  remain  in  existence,  the  Greenland  and 
Antarctic.  They  form  great  plateaus  sloping  upward  from  the 
margins;  the  Greenland  sheet  reaching  elevations  at  its  center 


THE  STRENGTH  OF  THE  EARTH'S  CRUST        39 

between  9,000  a?ad  10,000  feet,  the  Antarctic  attaining  to  about 
11,000  feet.  The  average  thickness  of  the  ice  must  be  thousands 
of  feet  in  each  case.  The  development  of  these  ice  caps  during 
the  refrigeration  of  climate  which  marked  the  later  Tertiary  must 
have  imposed  upon  the  crust  great  loads  of  wide  span.  If  isostatic 
equilibrium  was  previously  complete  to  a  large  degree,  the  ice 
mantles  should  give  valuable  measures  of  crustal  strength.  For 
this  purpose,  however,  a  set  of  gravity  measurements  should  be 
carried  inland  and  reduced  by  the  Hayfordian  method.  The  facts 
that  these  two  ice-mantled  areas  are  both  high  plateaus,  and 
that  no  other  adjacent  unglaciated  land  is  of  similar  topographic 
character,  suggest  that  these  regions  may  be  competent  to  carry 
great  thicknesses  of  ice  without  isostatic  yielding.  There  is  no 
present  basis,  however,  for  making  a  quantitative  estimate.  It 
must  be  borne  in  mind,  furthermore,  that  the  ice  mantle  is  only 
about  one-third  of  the  density  of  rock  and  that  lofty  mountains 
exist  in  both  regions,  showing  that  these  lands  would  possess  con- 
siderable mean  elevation  even  without  the  presence  of  the  ice. 
The  effect  of  the  difference  of  density  between  ice  and  rock  may 
be  appreciated  by  considering  that  an  ice  sheet  3,000  feet  in  thick- 
ness would  possess  the  same  mass  as  a  layer  of  rock  1,000  feet  thick. 
For  isostasy  to  remain  perfect  after  the  development  of  this  ice 
sheet,  the  crust  would  have  to  sink  1,000  feet,  but  the  surface  of 
the  ice  would  still  be  2,000  feet  above  the  former  level  and  give 
an  appearance  of  load  which  would  not  in  reality  exist. 

This  is  a  problem  meriting  research  for  several  reasons.  A 
knowledge  of  the  load  which  is  sustained  by  these  regions  would 
show  to  what  degree  the  warpings  connected  with  the  extinct 
Pleistocene  ice  sheets  were  mere  elastic  responses  to  load,  to 
what  degree  they  marked  subcrustal  plastic  flow  working  toward 
isostasy.  The  results  could  be  applied  also  to  the  problem  as  to 
how  far  from  isostatic  equilibrium  a  continent  might  come  to  lie 
as  a  result  of  continent-wide  base-leveling  in  a  period  of  geologic 
quiet.  It  seems  not  impossible  that  the  stress-curve  due  to  the 
portion  of  the  glacial  load  which  is  elastically  sustained  would  give 
stress-differences  greater  at  a  depth  of  300-500  km.  than  those 
shown  by  curve  C,  Fig.  18.  Such  an  investigation  may  then 


40  JOSEPH  BARRELL 

be  an  essential  factor  in  measuring  the  maximum  strength  of  the 
lithosphere  and  more  especially  the  asthenosphere. 

Accordance  of  geologic  with  geodetic  evidence. — The  United 
States  and  its  bordering  ocean  bottoms  is  a  region  of  moderate 
reliefs  as  compared  to  the  great  folds  of  the  ocean  floor  or  of  the 
continent  of  Eurasia.  The  geologic  forces  of  folding  and  uplift 
have  not  worked  here  with  their  greatest  intensity  and  the  central 
and  eastern  half  of  the  continent  has  been  affected  by  the  world- 
involving  Cenozoic  diastrophism  to  only  a  moderate  degree.  It 
is  to  be  expected  then  that  the  greatest  strains  upon  the  crust, 
the  maximum  departures  from  isostasy,  would  not  be  found  here. 
In  accordance  with  this  expectation  it  has  been  seen  that  by 
far  the  greatest  New  Method  gravity  anomalies  are  found  in 
other  regions  and  associated  in  most  cases  with  the  greater 
reliefs  of  the  globe.  The  geologic  evidence  is  in  harmony;  the 
amount  of  uncompensated  relief,  parallel  to  the  geodetic  evi- 
dence, is  greatest  for  the  lesser  wave-lengths;  but,  throughout,  the 
geologic  evidence  suggests  that  the  actual  burdens  which  can  be 
borne  by  the  crust,  as  found  in  regions  of  culminating  stress,  are 
appreciably  greater  than  those  detected  by  geodetic  methods  as 
existing  in  the  region  of  the  United  States. 

If,  in  some  past  ages,  as  during  the  Appalachian  or  Sierran 
revolutions,  strains  were  generated  in  this  continent  as  great  as 
those  found  now  in  some  other  regions,  it  would  appear  that  the 
slow  changes  of  geologic  time,  of  erosion  and  crustal  readjustment, 
have  partially  eased  the  crust  of  its  load.  We  may  have,  then,  a 
variable  crustal  strength — a  maximum  strength  exhibited  during 
and  following  the  crises  of  great  diastrophism;  another,  lesser 
strength,  which  measures  the  loads  which  the  crust  without  failing 
can  bear  through  all  of  geologic  time. 

ADJUSTMENT    OF    LOADS    TO    THE    DISTRIBUTION    OF    STRENGTH 

It  has  been  seen  that  the  departures  from  flotational  equilibrium 
may  become  very  notable  and  are  of  greatest  vertical  magnitude 
'or  wave-lengths  from  100  up  to  400  km.  The  strains  generated 
by  these  loads,  if  distributed  through  an  elastic  crust,consequently 


THE  STRENGTH  OF  THE  EARTH'S  CRUST        41 

reach  maximum  values  at  depths  not  exceeding  64  km.  Is  this 
because  the  earth  shell  below  the  zone  of  compensation  is  strong, 
but  for  some  unrelated  reason  free  from  large  stress-differences,  or 
is  there  an  absence  of  such  stress-differences  because  this  shell  is 
too  weak  to  bear  them?  If  the  latter  is  true,  then  the  relations 
of  amplitude  to  wave-length  which  have  been  developed  in  this 
chapter  offer  additional  proofs  of  the  reality  of  the  existence  of  the 
asthenosphere. 

The  geologic  evidence  on  the  evolution  of  continental  structures- 
and  elevations  leads  to  the  conclusion  that  the  distribution  of 
stress-differences  must  be  in  reality  the  result  of  the  existence  of  a 
zone  which  cannot  carry  large  distortional  strains,  as  may  be  seen 
upon  brief  consideration. 

The  internal  activities  of  igneous  intrusion  and  of  tangential 
compression  do  not  in  themselves  work  toward  isostatic  equilibrium, 
but  merely  toward  accentuation  of  relief.  Erosion  and  sedimenta- 
tion, while  tending  to  destroy  this  relief,  are  not  agents  tending  to 
create,  but  to  destroy,  such  isostatic  relations  as  have  developed. 
All  of  these  activities  work  on  a  continental  or  subcontinental  as 
well  as  on  an  orogenic  scale,  as  seen  in  the  Cenozoic  history  of  the 
broad  Cordilleran  province,  yet  while  the  orogenic  departures  from 
isostasy  are  vertically  very  great,  the  continental  departures  are 
very  moderate.  For  the  latter  there  must  be  then  some  more 
narrowly  limiting  condition.  This  corresponds  to  the  incapacity  of 
a  deep  zone,  the  asthenosphere,  to  carry  large  stress-differences  and 
the  incapacity  of  the  lithosphere  in  spite  of  its  greater  strength  to 
act  effectively  after  the  fashion  of  a  beam  for  loads  of  great  span. 
The  orogenic  structures,  on  the  other  hand,  give  maximum  stresses 
much  nearer  the  surface,  in  the  stronger  lithosphere;  because  of 
their  shorter  wave-lengths  they  do  not  produce  in  it  bending 
stress  as  in  a  loaded  beam  and  affect  comparatively  little  the 
deeper-seated  asthenosphere. 

If,  then,  it  is  known  from  the  preceding  theoretical  considera- 
tions that  the  limits  of  strength  of  the  lithosphere  and  astheno- 
sphere determine  the  limits  of  the  departures  from  isostasy,  the 
analysis  of  the  nature  of  these  departures  may  lead  in  turn  to  a 
knowledge  of  the  distribution  of  strength. 


42  JOSEPH  BARRELL 

CHARACTER   OF    THE    CURVE    OF    STRENGTH 

In  curve  F'  of  Fig.  18  is  shown  the  nature  of  the  curve  of 
strength  as  suggested  by  the  geodetic  evidence  from  the  United 
States.  In  curve  F"  is  shown  the  nature  of  the  curve  as  suggested 
by  the  departures  from  isostasy  exhibited  by  the  great  mountain 
axes  and  possibly  by  the  continental  ice  sheets.  These  curves 
may  be  taken  as  showing  the  value  of  the  elastic  limit  at  various 
depths  for  permanent  stresses.  With  varying  geologic  conditions, 
especially  those  connected  with  rising  magmas  and  their  emana- 
tions, the  curve  of  strength  must  vary  widely,  and  furthermore  no 
very  close  parallelism  of  strength-curve  and  stress-curve  is  to  be 
expected.  These  curves,  therefore,  are  intended  to  bring  out 
general  relations;  they  are  of  qualitative,  not  quantitative  value. 
The  drawing  of  curve  F"  somewhat  inside  of  curve  E  means  that 
below  the  point  of  maximum  stress  in  E,  as  given  for  a  homogeneous 
elastic  earth,  the  stress  is  assumed  as  somewhat  greater  than  the 
crust  at  those  levels  can  sustain.  Upon  the  development  of  this 
load  plastic  flow  in  these  deeper  levels  would  take  place  main- 
taining the  stress  within  the  strength  curve  for  each  level;  the 
crust  above  would  come  to  act  to  some  extent  as  a  bending  plate, 
the  stresses  within  it  would  increase,  chiefly  within  the  upper  and 
lower  portions.  This  added  strain  would  compensate  for  the  yield- 
ing below.  For  the  reasons  discussed  previously,  however,  show- 
ing the  structural  weakness  of  the  lithosphere  as  a  beam,  this  action, 
it  is  thought,  could  not  go  very  far,  and,  in  consequence,  the  loads 
on  the  lithosphere  are  essentially  such  as  to  give  stresses  contained 
within  it,  distributed  according  to  Darwin's  law.  The  preceding 
deals  only  with  that  part  of  the  curve  of  strength  which  marks  the 
gradation  from  lithosphere  to  centrosphere.  The  relations  of  this 
part  to  those  'above  and  below  need  still  to  be  considered. 

The  highest  stress  found  for  the  loads  regarded  as  harmonic 
waves  was  for  the  great  folds  on  the  floor  of  the  Pacific  Ocean. 
These  were  taken  as  equivalent  to  harmonic  waves  of  rock  of 
density  2.67,  400  km.  in  wave-length,  and  4,000  m.  in  amplitude. 
But  even  these  folds  give  a  maximum  stress  of  only  393  kg. 
per  sq.  cm.  (5,590  pounds  per  square  inch),  and  this  at  a 
depth  of  64  km.  At  the  surface  strong  limestone  or  granite  can 


THE  STRENGTH  OF  THE  EARTH'S  CRUST        43 

sustain  a  stress-difference  of  1,750  kg.  per  sq.  cm.  (25,000  pounds 
per  square  inch),  and  selected  specimens  show  ultimate  breaking 
strength  approaching  2,800  kg.  per  sq.  cm.  For  stresses  of  geologic 
endurance  and  in  the  heterogeneous  outer  crust  it  is  probable,  how^ 
ever,  that  stress  limits  should  be  chosen  below  1,750  kg.  per  sq.  cm. 

The  work  of  Adams  and  King  has  shown  that  small  cavities 
in  granite  are  not  closed  when  the  rock  is  subjected  to  the  pressure 
and  temperature  normally  existing  in  the  earth  at  a  depth  of  n 
miles.1  The  presence  of  occluded  gases  acting  through  great 
lengths  of  time,  by  facilitating  recrystallization,  might  affect  this 
result  of  laboratory  experiments,  but  the  capacity  of  dry  rock  to 
sustain  even  greater  cubic  pressures  without  yielding  seems  to  make 
safe  the  conclusion  that  except  in  the  presence  of  magmatic  emana- 
tions the  crust  at  a  depth  of  n  miles  (17.7  km.)  is  able  to  bear 
a  stress  difference  of  100,000  pounds  per  square  inch  and  is  at 
least  four  times  as  strong  as  rock  close  to  the  surface. 

At  twice  this  depth,  however,  the  temperatures  become  such 
that  if  it  were  not  for  the  great  pressures  even  dry  rocks  would 
approach  a  molten  condition.  The  presence  of  high  temperatures 
and  of  gases  which  may  act  as  crystallizers  presumably  becomes 
dominant  at  such  depths  over  the  effects  of  the  increasing  pressures. 
We  many  conclude,  therefore,  that  the  maximum  strength  of  the 
crust  in  regions  free  from  igneous  activity  is  found  at  levels  above 
rather  than  below  40  km.  and  may  lie  between  20  and  30  km.  deep. 

To  bring  to  a  focus  this  discussion  a  tabulation  of  ratios  of 
strengths  for  increasing  depths  may  be  given,  as  derived  from 
the  strength  curves  F',  F"  of  Fig.  18,  the  standard  being  taken  as 
the  strength  of  surface  rocks.  By  giving  them  merely  as  ratios 
and  stating  that  the  average  strength  of  the  solid  rocks  at  the. 
surface  is  itself  an  uncertain  quantity  owing  to  complications  of 
structure  and  composition,  the  appearance  of  an  undue  certainty  is 
avoided. 

The  general  conclusion  which  stands  out  from  this  tabulation  is 
that  the  weakest  part  of  the  asthenosphere  is  of  the  order  of  one 
one-hundredth  of  the  maximum  strength  of  the  lithosphere  and  is 
perhaps  only  a  twenty-fifth  of  that  of  massive  surface  rocks.  Its 

1  Jour.  GeoL,  XX  (1912),  97-138. 


44  JOSEPH  BARRELL 

limit  of  capacity  for  sustaining  stress-differences  is  apparently  of  the 
order  of  1,000  pounds  per  square  inch,  though  its  weakness  may  be 
masked  to  some  extent  by  the  strength  above.  From  the  evi- 
dence, however,  it  seems  capable  of  carrying  stresses  of  more  than 
100  pounds  per  square  inch,  but  is  clearly  incapable  of  carrying 
stresses  of  as  much  as  5,000  pounds  per  square  inch.  To  reach  a 

TABLE  XXX 

ESTIMATED   APPROXIMATE   RATIOS   GIVING  THE   VARIATION   OF   STRENGTH 
WITH  DEPTH  AS  SHOWN  BY  THE  NATURE  OF  DEPARTURES  FROM 

ISOSTASY 

LITHOSPHERE 

Depth  in  Kilometers  Strength  in  Percentage 

O  100 

20  4OO 

25  500 

30  400 

50  25 

ioo  17 

ASTHENOSPHERE 

Depth  in  Kilometers  Strength  in  Percentage 

200  8 

300  5 

400  4 

more  definite  conclusion  the  subject  must  be  tested  from  many 
angles  and  is  a  problem  for  the  geophysicist  rather  than  for  the 
geologist,  but  the  results  are  of  geological  importance  and  the 
geologic  and  geodetic  data  may  turn  out  to  have  more  determina- 
tive value  on  the  distribution  of  strength  than  the  evidence  from 
tides  and  earthquakes. 

[To  be  continued} 


THE  STRENGTH  OF  THE  EARTH'S  CRUST 


JOSEPH  BARRELL 
New  Haven,  Connecticut 


PART  VIII.    PHYSICAL  CONDITIONS  CONTROLLING  THE 
NATURE  OF  LITHOSPHERE  AND  ASTHENOSPHERE 

INTRODUCTION  AND  SUMMARY 425 

SECTION  A1 

RELATIONS  BETWEEN  RIGIDITY,  STRENGTH,  AND  IGNEOUS 

ACTIVITY 

DISTINCTIONS  IN  PHYSICAL  PROPERTIES  RELATED  TO  STRENGTH      .       .  429 
CONDITIONS  FAVORING  ASSOCIATION  OF  HIGH  RIGIDITY  WITH  Low 

ELASTIC  LIMIT 432 

ANALOGIES  BETWEEN  ASTHENOSPHERIC  ROCK  AND  GLACIAL  ICE      .       .  438 

RELATIONS  or  IGNEOUS  ACTIVITY  TO  ASTHENOSPHERE  AND  LITHOSPHERE  441 

INTRODUCTION  AND   SUMMARY 

The  experiments  of  F.  D.  Adams  have  demonstrated  that 
under  combined  pressures  and  temperatures  equal  to  those  existing 
at  a  depth  of  eleven  miles  granite  is  about  seven  times  stronger 
than  at  the  surface,  showing  that,  up  to  at  least  certain  limits,  the 
strength  of  the  crust  increases  downward.  The  measurements  of 
tidal  deformation  of  the  earth  and  of  the  variations  of  latitude 
concur  furthermore,  in  proving  that  the  rigidity  of  the  earth  as  a 
whole  is  greater  than  that  of  steel.  The  transmission  of  earth- 
quake vibrations  of  a  transverse  nature  through  the  earth  shows, 
not  only  that  the  earth  is  solid  and  rigid  throughout,  but  that,  on  the 
whole,  rigidity  increases  with  depth.  In  none  of  these  lines  of 
investigation  is  there  any  clear  suggestion  of  the  existence  of  a 
thick  shell  of  weakness — an  asthenosphere. 

On  the  other  hand,  the  conclusion  that  broad  areas  of  the 
crust  rest  in  approximate  isostatic  equilibrium  seems  to  imply  the 

1  Section  B  of  Part  VIII,  on  "Relations  with  Other  Fields  of  Geophysics,"  will 
be  published  in  the  succeeding  number  of  this  Journal. 

425 


426  JOSEPH  BARRELL 

existence  of  a  subcrustal  zone  with  but  little  strength,  readily 
yielding  under  vertical  loads  when  these  are  of  such  breadth  that 
the  strains  resulting  from  them  are  not  confined  and  absorbed 
within  the  strong  outer  crust. 

There  has  thus  been  developed  a  paradox,  an  apparent  conflict 
of  evidence  which  becomes  more  insistent  of  explanation  with  con- 
tinued accumulation  of  proofs  of  high  rigidity  from  the  domain  of 
geophysics  and  of  proofs  of  regional  isostasy  from  the  equally 
precise  field  of  geodesy. 

In  the  consideration  of  such  broad  problems,  the  hope  of  an 
ultimate  definitive  solution  rests  upon  the  use  of  the  method  of 
multiple  working  hypotheses.  The  surety,  significance,  and 
breadth  of  application  of  the  facts  must  be  established.  By  these 
the  various  hypotheses  must  be  tested  and  molded.  All  hypotheses 
must  be  kept  for  further  consideration  provided  they  are  not  posi- 
tively excluded.  In  the  complexity  of  relationships  there  is  com- 
monly a  complexity  of  cause,  and  hypotheses  which  seem  at  first 
to  be  mutually  exclusive  may  be  found  to  co-operate  in  giving  a 
completer  explanation.  Those  which  originally  appeared  antago- 
nistic may  thus  come  to  be  seen  as  participating  and  dividing  the 
field  of  cause  between  them.  A  paradox  often  points  to  this  kind 
of  a  conclusion. 

To  pass  to  the  particular  problem  of  the  relations  of  isostasy  to 
the  physical  conditions  of  the  earth's  interior;  the  hypothesis  devel- 
oped in  this  study — of  the  existence  of  a  zone  of  weakness  under- 
lying a  zone  of  strength — must  not  be  regarded  at  present  as  the 
only  available  hypothesis.  Searching  investigation  must  be 
carried  forward  to  see  if  other  and  possibly  antagonistic  hypotheses 
cannot  be  developed  which  will  equally  well  co-ordinate  and  explain 
the  facts.  Even  if  true  in  the  main,  it  is  likely,  as  has  been  the  case 
with  other  hypotheses,  that  time  will  show  that  in  certain  directions 
it  has  been  carried  too  far.  Such  testing,  however,  can  best  be 
done  by  others,  and  after  the  implications  of  this  hypothesis  are 
seen.  In  this  part,  the  concluding  article  of  this  series,  a  discussion 
had  best  be  given  of  the  lines  of  adjustment  by  which  the  hypothesis 
here  favored  may  be  brought  into  harmony  with  other  fields  of 
geophysical  evidence.  With  this  understanding  of  the  relation  of 


THE  STRENGTH  OF  THE  EARTH'S  CRUST       427 

the  present  investigation  to  the  method  of  multiple  working  hy- 
potheses, examination  will  be  made  of  the  paradox  which  has  been 
drawn  between  certain  conceptions  from  other  lines  of  investiga- 
tion and  those  drawn  from  this  study  of  crustal  strength. 

Having  given  this  introductory  presentation  on  what  is  con- 
ceived to  be  a  judicial  point  of  view,  we  may  turn  to  a  review  of  the 
conclusions  reached  in  this  article.  It  is  pointed  out  that  rigidity 
is  strictly  a  measure  of  stiffness;  whereas  a  very  different  quality, 
the  limit  of  elastic  yielding,  or  the  beginning  of  flow,  is  the  measure 
of  strength.  But  mass  flowage  may  take  place  in  a  number  of  quite 
different  ways,  according  to  the  nature  of  the  solid  and  the  environ- 
ing physical  and  chemical  conditions.  The  elastic  limit  and  hence 
the  strength  will  differ  in  the  same  solid  according  to  the  mode  of 
yielding.  Four  modes  may  be  here  enumerated  in  what  is  thought 
to  be  their  order  of  increasing  importance,  the  fourth  mode  being 
that  which  is  conceived  as  operative  especially  in  the  asthenosphere, 
and  serving  to  maintain  the  condition  of  approximate  regional 
isostasy. 

First,  flowage  may  take  place  rapidly  by  true  plastic  or  molecular 
flow,  as  with  lead  or  white-hot  iron,  the  solid,  when  stressed  well 
beyond  the  elastic  limit,  behaving  like  a  viscous  fluid.  It  is  not 
thought  that  the  terrestrial  deformations  are  often  carried  on  with 
a  rapidity  which  requires  true  plastic  yielding.  In  fact,  under  such 
rapid  stresses  as  those  produced  by  earthquakes  and  tides  it  is  not 
improbable  that  the  strength  of  the  earth  may  progressively 
increase  with  depth. 

As  a  second  and  quite  different  mode,  deformation  may  take 
place  by  molar  as  distinct  from  molecular  shear.  In  the  zone  of 
fracture  this  is  manifested  in  jointing  and  faulting  and  is  empha- 
sized as  distinct  from  rock  flowage,  but  where  the  fracturing 
becomes  so  closely  spaced  as  to  result  in  slicing  of  individual 
minerals  it  passes  under  the  category  of  granulation.  Where 
carried  on  at  depth  there  is  always  some  degree  of  cementation  by 
recrystallization.  Deformation  by  such  close-grained  fracture 
without  complete  loss  of  cohesion  is  classed  as  rock  flowage.  It 
is  thought  to  be  developed  to  some  degree  within  the  lithosphere, 
especially  by  great  horizontally  compressive  forces,  but  is  not 


428  JOSEPH  BARRELL 

regarded  here  as  the  mode  of  deep  rock  flowage  involved  in  the 
isostatic  readjustment  of  unfolded  tracts. 

Thirdly,  flowage  may  take  place  in  some  minerals,  as  calcite 
and  ice,  by  gliding  upon  the  cleavage  planes.  But  such  gliding  is 
not  regarded  as  the  mode  by  which  the  foliated  rocks  are  developed. 
It  requires  furthermore  far  greater  force  than  that  which  is  given 
by  the  departures  from  isostatic  equilibrium. 

A  fourth  mode  of  rock  flowage  is  by  recrystallization.  It  is  the 
chief  factor,  as  Van  Hise  has  shown,  in  the  deformation  of  the 
crystalline  foliates.  It  is  thought  that  this  is  also  the  method  by 
which  the  asthenosphere  yields  and  that  a  readiness  of  recrystalli- 
zation under  unbalanced  stresses  of  a  permanent  nature  is  the  cause 
of  the  weakness  of  the  asthenosphere. 

The  vibratory  forces  transmitted  as  earthquake  shocks  and  those 
due  to  tidal  strain,  from  this  standpoint,  are  both  rapid.  Under 
such  conditions  the  asthenosphere  could  show  high  order  of  strength. 
It  is  argued  that  the  ease  of  recrystallization  under  constant  strain 
becomes  more  marked  the  nearer  the  temperature  approaches  to 
that  of  fusion,  or  to  express  it  better  from  the  physico-chemical 
standpoint,  the  nearer  the  temperature  approaches  to  the  mutual 
solution  point  of  the  cons titu tents  involved.  The  result  is  that  at 
such  temperatures  the  rigidity  may  be  high  and  not  greatly  different 
from  that  at  low  temperatures,  but  for  permanent  stresses  the 
elastic  limit  becomes  low.  The  movement  of  continental  glaciers 
with  a  low  surface  gradient,  accomplished  by  recrystallization, 
illustrates  the  condition  which  it  would  appear  exists  to  even  a 
higher  degree  within  the  asthenosphere. 

This  conclusion  carries  with  it  the  idea  that  within  the  litho- 
sphere  the  temperature  is  in  general  considerably  below  that  of 
fusion;  whereas  below,  in  the  thick  zone  of  weakness,  the  tempera- 
ture must  lie  close  to,  or  at,  that  of  molten  rocks.  For  fusion  there 
is  needed,  however,  the  energy  necessary  to  supply  the  latent  heat 
and  volume  expansion.  Unless  this  energy  is  supplied,  the  astheno- 
sphere remains  solid  rock,  but  the  least  accession  of  internal  heat, 
or  relief  from  external  pressure,  will  generate  a  proportionate 
amount  of  magma,  diffused  as  liquid  throughout  the  solid.  To 
gather  into  reservoirs  temporarily  molten,  the  magma  must  con- 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  429 

verge  by  rising,  analogous  to  the  draining  of  melted  water  from 
glaciers;  uniting,  as  rivulets  unite  into  rivers,  and  rivers  discharge 
into  lakes.  No  continuous  lava  stratum  or  large  reservoirs  of  lava 
could,  under  the  terms  of  this  hypothesis,  be  expected  to  exist 
within  the  asthenosphere.  Its  very  weakness  would  prevent  it 
from  acting  as  a  containing  vessel  for  holding  large  volumes  of  any 
fluid  which,  for  any  cause  such  as  a  lower  specific  gravity  of  the 
fluid  phase,  would  tend  to  rise.  The  evidence  of  earthquake  vibra- 
tions and  of  resistance  to  tidal  deformation  further  supports  the 
view  that  the  asthenosphere  is  not  a  liquid  or  even  a  truly  viscous 
zone.  On  the  other  hand,  only  in  the  lithosphere  would  be  found 
the  strength  needed  for  the  storage  of  magma  in  volumes  until  the 
limit  of  its  strength  as  a  containing  vessel  was  reached. 

Partly  guided  by  observation  upon  the  metamorphic  rocks, 
partly  by  theories  of  the  nature  of  deformation  at  great  depths, 
the  argument  leads  to  conclusions  on  the  mode  of  yielding  within 
the  different  levels  of  the  crust.  First,  the  outermost  zone  is 
observed  to  be  a  zone  of  fracture,  weak  in  comparison  with  the 
thick  zone  below.  This,  the  second  zone,  is  the  zone  of  strength  and 
yields  by  flowage,  but  flowage  which  is  characterized  by  granulation 
as  the  dominant,  by  recrystallization  as  the  subordinate,  mode. 
The  expenditure  of  energy  for  a  given  deformation  is  here  a  maxi- 
mum. In  the  third  zone,  the  asthenosphere,  on  the  contrary, 
flowage  is  conceived  as  taking  place  with  but  little  expenditure 
of  energy,  by  a  ready  recrystallization  at  the  temperature  of 
primary  crystallization  of  magmas.  Those  contorted  granite- 
gneisses  seen  especially  in  the  Archean  rocks,  which  are  regarded 
as  deformed  during  the  final  stages  of  crystallization,  exhibit 
locally  in  the  outer  crust  the  conditions  which  it  appears  may 
permanently  prevail  within  the  asthenosphere. 

SECTION  A 

RELATIONS  BETWEEN  RIGIDITY,  STRENGTH  AND  IGNEOUS 

ACTIVITY 

DISTINCTIONS  IN  PHYSICAL  PROPERTIES  RELATED  TO  STRENGTH 

Elasticity  is  of  two  natures:  that  of  volume  and  that  of  form. 
The  first  is  possessed  by  matter  in  either  the  gaseous,  liquid,  or 
solid  state;  the  second  is  possessed  by  solids  only  and  is  associated 


430  JOSEPH  BARRELL 

with  rigidity  and  strength.  Up  to  a  certain  degree  of  strain  known 
as  the  elastic  limit,  elasticity  of  form  in  the  ideal  solid  is  perfect  and 
is  expressed  by  the  law  that  the  change  of  form,  or  strain,  is  directly 
proportional  to  the  load  applied,  or  stress.  This  load  may  be 
maintained  indefinitely  and,  except  for  a  slight  relaxation,  the  solid 
shows  no  further  yielding.  Upon  the  removal  of  the  load  there 
is  an  elastic  return  to  the  original  form,  but  the  very  last  stages 
of  the  recovery  are  slow.  The  elastic  nature  of  the  whole  earth, 
in  regard  to  both  volume  and  form,  is  shown  by  its  capacity  to  trans- 
mit the  several  kinds  of  earthquake  vibrations.  The  permanence 
and  perfection  of  the  elasticity  of  form  is  also  implied  in  the  power 
of  the  crust  to  carry  loads  up  to  certain  limits  for  times  reaching  into 
geologic  periods  without  exhibiting  progressive  viscous  yielding. 

Beyond  the  elastic  limit,  elasticity  ceases  to  be  perfect,  and  a 
permanent  change  of  form  occurs.  This  relieves  part  of  the  stress 
and  reduces  the  strain  to  within  the  elastic  limit.  The  change  of 
form  may  be  by  rupture,  in  which  event  the  strength  of  the  body 
is  destroyed.  It  may  be  by  plastic  flow,  in  which  case  the  strength 
may  be  increased  or  decreased.  Wrought  iron,  for  example, 
becomes  somewhat  stronger  as  a  result  of  forging.  Granite  on 
being  mashed  into  gneiss  becomes  somewhat  weaker  because  of 
the  development  of  weaker  minerals,  especially  the  micas.  In  the 
crust  of  the  earth,  except  for  the  outer  few  miles,  flowage  takes 
place  without  probably  much  change  in  the  mineral  composition 
and  consequently  in  the  strength  of  the  rock.  Deformation  will 
continue  as  long  as  the  stress  is  maintained  well  above  the  elastic 
limit,  but  upon  the  cessation  of  movement  there  may  still  remain 
residual  stresses  up  to  the  elastic  limit.  If  the  residual  stresses 
over  broad  areas  are  small,  it  may  be  because  the  development  of 
weaker  structures,  such  as  folds  or  zones  of  igneous  injection,  has 
eased  the  strains. 

Failure  by  flow  brings  in  the  distinction  between  viscosity  and 
plasticity.  These  are  often  used,  even  by  physical  geologists, 
as  merely  synonymous  terms,  but  there  is  a  real  distinction  which 
should  be  noted.  Fluids  are  viscous  to  a  small  or  large  degree  and 
can  have  no  elasticity  of  form.  Viscous  flow  must,  however,  over- 
come internal  friction  and  requires  time  for  its  accomplishment. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  431 

With  long  time  even  a  minute  force  will  cause  even  a  very  viscous 
fluid  to  flow.  Solids,  on  the  other  hand,  possess  elasticity  of  form, 
and  below  the  elastic  limit  can  hold  shearing  stresses  indefinitely. 
Above  it  they  may  flow  and  in  so  doing  exhibit  plasticity.  The 
phenomenon  differs  from  viscosity  in  that  the  force  must  rise  to  a 
certain  magnitude  before  any  gliding  between  molecules  begins. 
The  crust,  then,  is  plastic  but  not  viscous. 

Although  the  theoretical  distinction  between  plasticity  and  vis- 
cosity is  clear,  recognition  must  be  given  to  conditions  where  the 
two  states  merge.  This  is  especially  true  for  undercooled  glasses. 
A  glass  in  its  molecular  organization  is  a  liquid  and  yet  it  possesses 
definite  elastic  moduli  and  elastic  limit.  From  this  standpoint  of 
elasticity  the  glass,  therefore,  is  a  solid.  Upon  rise  of  temperature 
there  is,  however,  no  absorption  of  latent  heat  to  mark  a  change  of 
state,  the  elastic  limit  gradually  lowers,  disappears  for  prolonged 
stresses,  and  elastically,  the  substance  passes  by  gradation  from  a 
solid  to  a  liquid.  The  existence  of  these  transition  cases  should  not, 
however,  be  permitted  to  obscure  the  real  distinctions  between 
solid  and  liquid. 

The  crust  yields  as  a  plastic  solid  to  forces  which  strain  it 
beyond  its  elastic  limit.  But  the  solid  flowage  which  this  implies 
may  be  either  by  distortion  of  crystals  or  by  recrystallization. 
The  first  is  familiar  for  rapidly  applied  forces,  requires  compara- 
tively great  stress,  and  corresponds  to  the  usual  conception  of 
plasticity.  The  crystalline  rocks  make  us  familiar,  however,  with 
the  idea  of  mass  plasticity  by  recrystallization.  This  is  plasticity ? 
but  in  a  somewhat  different  sense  from  that  which  is  usually  con- 
veyed by  the  term. 

The  degree  of  elasticity  which  a  substance  may  exhibit  is  a 
different  property  from  the  elastic  limit.  A  bar  of  wrought  iron 
one  square  inch  in  cross-section  will  be  elongated  one  part  in  28,000,- 
ooo  by  a  tensile  stress  of  one  pound.  A  similar  bar  of  glass  would  be 
elongated  one  part  in  10,500,000,  more  than  twice  as  much.  These 
ratios  measure  the  degree  of  elasticity  under  tensile  or  compressive 
stresses  and  differ  for  each  substance.  The  figure  is  known  as 
Young's  modulus  of  elasticity.  A  substance  may  be  highly  elastic, 
that  is,  have  a  high  modulus  of  elasticity,  as  cast  iron,  or  glass, 


432  JOSEPH  BARRELL 

and  yet  be  brittle  because  of  a  low  elastic  limit  under  rapid  tensile 
stress,  combined  with  lack  of  plasticity  at  ordinary  temperatures. 
At  temperatures  sufficiently  high,  the  modulus  is  not  greatly 
different,  but  the  elastic  limit  is  still  lower.  The  substance  is  now, 
however,  plastic,  rather  than  brittle,  since  plasticity  is  greatly 
increased;  but  a  rapid  strain,  exceeding  the  rapidity  with  which 
plastic  deformation  can  take  place,  may  still  produce  fracture. 
Another  substance,  such  as  rubber,  may  have  a  low  modulus  of 
elasticity  and  yet  a  relatively  high  elastic  limit. 

Among  similar  substances  under  similar  physical  conditions  there 
is,  however,  a  definite  association  of  these  properties  which  for  the 
metals  is  brought  out  well  in  a  tabulation  by  Johnston  and  L.  H. 
Adams.1  It  is  shown  for  a  class  of  substances,  such  as  the  metals, 
that  the  modulus  of  elasticity,  the  hardness,  the  tensile  strength,  and 
the  elastic  limit  all,  so  far  as  the  data  are  given,  occur  in  the  same 
order;  so  that  of  two  metals  that  which  has  the  higher  elastic  limit 
is  the  higher  also  in  the  other  qualities.  From  this  association 
there  results  a  ready  mental  confusion  between  rigidity  and 
strength.  The  one,  however,  denotes  the  degree  of  resistance  to 
distortion  from  a  unit-shearing  stress  and  gives  the  modulus  of 
ridigity.  The  other  is  measured  by  the  elastic  limit.  As  an 
example  of  the  confusion  between  these  two  different  properties, 
it  is  known  that  the  earth  as  a  whole  is  more  rigid  than  steel.  This 
to  many  would  appear  to  mean  that  it  was  stronger  than  steel. 
Earthquake  waves  show  that  the  earth  becomes  progressively 
more  incompressible  and  more  rigid  with  depth.  This  might  be 
held  as  evidence  against  the  existence  of  a  thick  sphere  of  weakness, 
the  asthenosphere.  High  incompressibility  and  high  rigidity  are 
not,  however,  direct  testimony  of  strength,  and  it  is  the  purpose 
of  the  next  topic  to  show  under  what  conditions  a  solid  may  be  very 
rigid  and  yet  very  weak. 

CONDITIONS   FAVORING  ASSOCIATION   OF   HIGH  RIGIDITY   WITH  LOW 

ELASTIC  LIMIT 

Alpine  glaciers  as  well  as  the  Alpine-like  margins  of  the  Green- 
land ice  sheet  move  much  more  rapidly  in  the  summer  than  in  the 

1  "On  the  Effect  of  High  Pressures  on  the  Physical  and  Chemical  Behavior  of 
Solids,"  Amer.  Jour.  Sci.,  XXXV  (1913),  220. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  433 

winter,  a  phenomenon  to  be  accounted  for  by  the  rate  of  recrystal- 
lization.  The  parts  of  an  ice  crystal  which  are  subjected  to  shear 
and  compression  have  the  melting-point  lowered.  They  melt,  dis- 
charge the  strain,  and  refreeze.  In  the  winter  the  general  tempera- 
ture is  reduced,  and  a  greater  strain  is  necessary  to  bring  the 
melting-point  down  to  the  lower  temperature.  Until  local  melting 
is  produced  the  ice  behaves  like  any  other  crystalline  solid,  as 
a  substance  possessing  elasticity  of  form.  Beyond  that  point  it 
exhibits  plasticity  and  behaves  in  some  respects  like  a  very  viscous 
fluid.  In  other  respects,  however,  it  exhibits  properties  quite 
distinct  from  that  of  the  usual  conception  of  mere  plastic  flow,  since 
in  the  testing  machine,  or  on  the  walls  of  a  crevasse,  ice  will  resist 
strong  shearing  strains,  and  yet  the  glacier  as  a  whole  yield  s  an 
flows  slowly  under  a  moderate  pressure-difference  as  shown  by  the 
low  gradient  of  its  upper  surface.  Glacial  motion  appears  to  take 
place,  therefore,  by  the  solution  and  growth  of  crystals,  not  by  a 
true  viscous  flow.  The  solid  and  crystalline  nature  throughout  as 
opposed  to  viscous  fluidity  is  furthermore  shown,  as  Chamberlin 
has  noted,  in  the  power  of  the  glacial  ice  to  shove  over  and  abrade 
its  floor  and  to  ride  up  slopes.  Chamberlin  adds  that  a  dry  glacier 
is  a  rigid  glacier.  A  dry  glacier  is  necessarily  cold,  and  a  cold 
glacier  is  necessarily  dry.1 

With  ice  subjected  to  slowly  applied  forces  the  elastic  limit 
is  consequently  dependent  upon  the  point  of  yielding  by  recrystalli- 
zation.  We  thus  see  an  intimate  relationship  between  tempera- 
ture and  variation  in  the  elastic  limit,  the  elastic  limit  for  ice  being 
greater  for  low  temperatures  than  for  high  temperatures.  But  the 
modulus  of  rigidity,  on  the  contrary,  measures  the  elastic  change 
of  form  for  unit-shearing  force,  change  of  form  not  accompanied 
by  crystallization,  but  marked  by  a  capacity  to  spring  back  to  the 
original  form  upon  the  removal  of  the  stress. 

T.  W.  Richards,  in  his  studies  on  the  compressibility  of  solids, 
notes  that  they  are  almost  as  compressible  and  voluminous  at 
absolute  zero  as  at  ordinary  temperatures.  Under  this  conception 

X"A  Contribution  to  the  Theory  of  Glacial  Motion,"  Decennial  Publications 
of  the  University  of  Chicago,  IX,  203,  204  (1904);  Chamberlin  and  Salisbury,  Geology, 
I  (1904),  305. 


434  JOSEPH  BARRELL 

the  molecules,  taken  as  equivalent  to  their  spheres  of  influence,  are 
in  actual  contact  and  suffer  mutual  compression  owing  to  the 
attraction  of  cohesion.  The  influence  of  heat  is  relatively  unim- 
portant in  determining  the  density  of  a  solid.  The  atoms  are  in 
most  cases  even  more  compressed  and  distorted  by  the  converging 
force  of  chemical  affinity  than  are  the  molecules  by  cohesion.  This 
corresponds  with  the  fact  that  substances  of  small  atomic  volume 
are  on  the  whole  more  incompressible  than  those  of  greater 
atomic  volume.  For  the  more  incompressible  substances  also 
the  decrease  in  compressibility  with  added  load  is  relatively  little, 
suggesting  that  they  are  already  greatly  compressed  by  the  forces 
of  chemical  affinity.1  The  effect  of  heat  serves  only  to  distend 
slightly  the  spheres  of  influence  of  the  atoms,  so  long  as  the  sub- 
stance is  in  the  solid  state.  Rise  of  temperature  to  near  the 
melting-point,  as  long  as  there  is  not  a  softening  by  the  develop- 
ment of  incipient  liquidity,  should,  according  to  these  views, 
change  the  elastic  properties  but  slightly.  The  greatly  lessened 
strength  of  ice  near  the  melting-point,  as  expressed  in  the  freedom 
of  regelation,  is  not,  following  these  ideas,  connected  with  the 
slightly  lessened  incompressibility  and  rigidity.  For  many  sub- 
stances the  problem  is  complicated,  however,  by  changes  of  molecu- 
lar state  with  changes  in  temperature  and  pressure.  This  is 
especially  true  of  ice  when  subjected  to  extreme  ranges  in  tempera- 
ture and  pressure,  as  has  been  shown  by  Bridgeman;  for  ordinary 
glacial  ice,  however,  we  deal  with  but  a  single  state. 

For  ice  at  a  temperature  of  —  7?  03  C.  the  compressibility  has 
been  determined  for  pressures  ranging  approximately  between 
100  and  500  atmospheres.  It  is  found  to  possess,  according  to 
Richards  and  Speyers,  about  one-fourth  of  the  compressibility  of 
water  at  neighboring  temperatures  and  about  five  times  the  com- 
pressibility of  glass.2  But  glass  possesses  a  compressibility  between 
that  of  acidic  and  basic  holocrystalline  igneous  rocks.  Ice  may  be 
taken  then  as  about  three  or  four  times  as  compressible  as  granite. 

1  "The  Present  Aspect  of  the  Hypothesis  of  Compressible  Atoms,"  Am.  Chem.  Soc. 
Jour.,  XXXVI  (1914),  2417-39. 

2  T.  W.  Richards  and  C.  L.  Speyers,  "The  Compressibility  of  Ice,"  Am.  Chem.  Soc. 
Jour.,  XXXVI  (1914),  49*^94- 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  435 

Now  the  modulus  of  rigidity  is  related  to  the  modulus  of  com- 
pressibility by  means  of  a  formula  involving  Poisson's  ratio.1 
This  ratio  varies  for  each  substance,  but  for  rocks,  for  iron  and 
steel,  and  probably  for  ice,  it  lies  between  o .  2  and  o .  3  in  value,  so 
that  in  general  the  rigidity  of  these  substances  can  be  judged  roughly 
by  their  degree  of  incompressibility.  Consequently  it  is  seen  that 
glacier  ice  at  temperatures  such  as  those  which  prevail  in  the  body 
of  the  moving  glacier  possesses  a  degree  of  incompressibility  and 
rigidity  which,  if  these  elastic  constants  were  measures  of  its 
strength,  would  make  it  wholly  incapable  of  motion  on  such 
gradients  as  are  observed.  This  can  be  made  more  obvious  by 
some  quantitative  statements.  Granite  and  similar  rocks,  for 
example,  can  stand  permanently  in  steep  cliffs  to  heights  of  thou- 
sands of  feet.  They  constitute  mountain  ranges  whose  height 
and  steepness  are  limited  entirely  by  the  forces  of  erosion  on  the 
one  hand  and  the  strength  of  the  asthenosphere  on  the  other.  The 
cliffs  could  be  very  much  higher  and  the  mountains  much  more 
lofty  before  glacier-like  flow  at  the  base  of  the  mountain  mass  would 
occur.  In  fact,  with  a  compressive  strength  of  25,000  pounds  per 
square  inch,  a  rectangular  block  of  granite  could  stand  as  a  vertical 
wall  22,000  feet  high,  and  of  indefinite  breadth,  without  yielding 
of  the  base.  With  a  sloping  face  and  supported  by  spurs  such  as 
occur  in  nature,  the  height  of  the  granite  mass  could  become  con- 
siderably greater.  For  parallel  mountain  ranges  of  harmonic  form 
and  gentle  slopes  resting  upon  a  foundation  whose  compressive 
strength  to  indefinite  depths  was  25,000  pounds  per  square  inch,  the 
mountain  crests  could  stand  eleven  miles  above  the  valley  bottoms 
before  the  maximum  stress-difference  would  reach  this  limit. 
Even  then,  if  the  slopes  were  as  low  as  those  of  a  continental  ice 
sheet,  the  failure  would  not  take  place  by  flowage  of  the  mountains 
laterally  into  the  valleys,  but  by  a  vertical  settling  of  the  mountains 
and  a  vertical  upwarping  of  the  valleys.  The  lateral,  plastic  flow 
would  be  at  some  depth  in  the  earth.  If  the  asthenosphere  were 
indefinitely  rigid,  granite  mountains  of  sufficiently  gentle  slope 

1  Let  P  =  Poisson's  ratio;  C,  the  modulus  of  rigidity;  D,  the  modulus  of  com- 
pressibility. ThenC=-  *~2  D . 


436  JOSEPH  BARRELL 

could  rise  to  indefinite  heights.  This  is  because  the  depth  of 
maximum  stress-difference  would  lie  at  about  one-sixth  of  the 
wave-length  below  the  mean  level  of  the  surface.  With  increasing 
wave-length  the  height  of  the  waves  could  accordingly  be  greater 
without  increasing  the  stre^-difference  at  the  trough-line  of  the 
waves.  The  gradient  would,  however,  have  to  become  more 
gentle;  in  other  words,  the  amplitude  would  have  to  increase  at  a 
lesser  rate  than  the  wave-length.  If  the  strength  of  ice  were  meas- 
ured by  its  rigidity  it  could  stand  permanently  in  masses  one-third 
or  one-fourth  as  steep  and  high  as  these  theoretic  limits  for  granite 
mountains,  without  failure  by  plastic  flow.  Yet,  on  the  contrary, 
the  great  ice  fields  spread  out  by  flowage  of  their  bases,  although 
their  surfaces  possess  very  gentle  gradients.  The  distinction 
between  strength  and  rigidity  in  the  movement  of  glaciers  is  thus 
clear.  The  strength  of  glaciers  is  limited  by  the  amount  of  the 
stress-differences  needed  to  produce  slow  movement  by  recrystalli- 
zation. 

Johnston  and  L.  H.  Adams  have  applied  this  theory  of  yielding, 
well  known  as  an  explanation  of  glacial  motion,  to  all  plastic  flow, 
and  argue  that  even  for  those  substances,  such  as  the  metals  and 
rocks,  in  which  cubic  compressibility  raises  the  melting-point, 
shear  greatly  lowers  it  for  the  parts  under  stress.1  They  argue  from 
a  physico-chemical  basis  that  the  most  plausible  explanation  for 
flow  in  metals  is  that  the  shearing  strain  is  great  enough  on  individ- 
ual points  to  produce  a  change  of  phase  of  individual  molecules 
from  solid  to  liquid,  even  at  ordinary  temperatures. 

Apart  from  theory  as  to  its  explanation,  the  phenomenon  of 
welding  of  iron  shows  for  high  temperatures  a  low  elastic  limit  and 
ready  passage  beyond  into  plastic  flow.  For  iron  and  steel, 
furthermore,  the  influence  of  temperature  upon  the  rigidity  has 
been  investigated.  Pisati  gives  the  following  equations  in  which 
n  is  the  value  of  the  modulus  of  rigidity  for  temperature  t.2  For 
iron — 

Wf=8nXio6(i—  .000,2062—  .000,000, 1 9/2+.  000,000,001,  iP), 

1  "On  the  Effect  of  High  Pressures  on  the  Physical  and  Chemical  Behavior  of 
Solids,"  Am.  Jour.  Sci.,  XXXV  (1914),  205-53. 

2  Smithsonian  Physical  Tables  (1904),  p.  76. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  437 

for  steel— 

io6(i  —  .  ooo,i8y/—  .  000,000,59^+  .  ooo,ooo,ooo,9/3). 


This  equation  for  iron  gives  a  minimum  modulus  at  314°  C.  equal 
to  95  per  cent  of  the  modulus  at  o°^.  For  steel  the  minimum 
value  occurs  at  342°  C.  and  is  90  per  cent  of  the  modulus  at  zero. 
At  528  °C.  iron  has  the  same  modulus  as  at  o°  C.  and  at  890°  C.  steel 
has  the  same  modulus  as  at  zero.  Doubtless  1000°  C.  is  above  the 
limits  of  the  data  from  which  these  formulas  were  derived.  For 
this  temperature  they  may  consequently  give  inaccurate  results, 
but  it  is  of  interest  to  note  that  the  curve  gives  a  modulus  of 
rigidity  for  iron  at  that  temperature  i  .  7  times  that  at  o°  C.  and 
for  steel  i  .  i  times  that  at  o°  C.  The  extrapolation  prevents 
attaching  quantitative  value  to  these  figures,  but  the  qualitative 
conclusion  may  be  reached  that  iron  and  steel  at  high  temperatures 
do  not  exhibit  less  rigidity  than  they  possess  at  lower  temperatures. 
It  is  obvious,  however,  that  above  a  certain  temperature  the  elastic 
limit  becomes  very  low,  as  shown  by  the  capacity  for  forging, 
and  for  strains  beyond  this  limit  deformation  takes  place  by 
plastic  flow.  That  it  is  not  merely  incipient  fusion  is  suggested 
by  the  maintenance  of  a  crystalline  condition  through  the  process 
of  deformation.  The  subject  for  iron  is  doubtless  complicated  by 
the  fact  that  iron  passes  through  more  than  one  solid  molecular 
state  in  being  heated  up  to  fusion.  Presumably  then  the  equation 
given  for  the  relation  of  rigidity  to  temperature  can  only  be  a  first 
approximation  to  the  actual  changes. 

Let  the  attention  be  given  next  to  the  crystalline  rocks  which 
were  once  deep-seated  and,  owing  to  subjacent  batholithic  invasion, 
attained  their  crystallization  at  exalted  temperatures.  It  is  ob- 
served that,  although  the  rock  masses  have  been  extensively 
deformed,  the  individual  crystals  have  regrown  during  the  process 
so  as  to  possess  compact  boundaries,  and  an  internal  constitution 
nearly  free  from  strain.  The  interpretation  is  that  the  deforma- 
tions due  to  geologic  forces  were  so  slow  and  the  rocks  were  so 
saturated  with  crystallizing  agents  at  high  temperatures  that 
recrystallization  could  nearly  keep  pace  with  the  deformation,  even 
for  temperatures  below  the  range  of  plasticity.  As  understood  by 


438  .      JOSEPH  BARRELL 

the  students  of  anamorphism,  the  process  has  depended  upon  a 
readier  solution  of  the  molecules  under  shearing  stress  than  of  those 
free  from  such  stress — solution  carried  on  by  means  of  the  relatively 
minute  proportion  of  gaseous  crystallizers  which  were  present 
through  the  rock  mass.  Such  crystallizers  doubtless  facilitate  the 
process.  They  form,  in  fact,  solutions  with  the  rock  which  may  be 
regarded  as  mixtures  with  very  low  fusion  points.  But,  theoretic- 
ally, as  the  temperatures  approach  those  of  general  fusion  the  need 
of  such  crystallizers  diminishes.  Moderate  shearing  stresses  can 
thus  liquefy  the  parts  upon  which  they  act  and  a  process  analogous 
to  glacial  motion  sets  in  for  solid  rock.  The  strength  of  highly 
heated  rock  appears  then  dependent  upon  the  amount  by  which  the 
temperature  is  below  the  melting-point.  That  zone  of  the  earth 
which  is  very  weak  may  then  be  regarded  as  approximately  at  the 
temperature  of  fusion.  To  transform  the  solid  into  liquid  there  is 
needed  only  the  energy  required  for  latent  heat  and  increase  of 
volume.  The  proportion  of  liquid  which  is  generated  will  vary 
directly  with  the  amount  of  heat  supplied  or  the  amount  of  hydro- 
static pressure  removed.  Magma,  consequently,  can  be  generated 
in  this  zone  more  readily  than  above  in  the  zone  of  strength;  but 
it  will  not  be  in  reservoirs;  rather  will  it  be  in  its  place  of  origin 
disseminated  through  the  rock  mass  like  water  standing  in  a  porous 
sandstone.1 

ANALOGIES  BETWEEN  ASTHENOSPHERIC  ROCK  AND  GLACIAL  ICE 

The  theory  of  the  asthenosphere  as  here  presented  is  seen  to 
have  important  relations  to  other  branches  of  geology.  The  zone 
of  weakness  becomes  especially  the  generator  of  magmas;  the 

1  Recently  the  writer  has  learned  from  Mr.  Bailey  Willis  of  an  unpublished  paper 
which  he  gave  some  years  ago  to  the  Geological  Society  of  Washington,  in  which  he 
outlined  his  views  of  the  nature  of  crustal  thrusts  as  illustrated  by  the  Appalachians. 
In  that,  and  more  recently,  as  a  result  of  studies  in  the  Alps  and  Andes,  he  has  come  to 
entertain  the  view  that  the  zone  of  compensation,  the  lithosphere,  shears  over  the  zone 
below  through  the  agency  of  molecular  or  mass  fusion.  Deep-seated  horizontal  shear 
and  igneous  intrusion  he  thus  holds  have  important  associations  with  erogenic  move- 
ments. We  have  thus  arrived  independently  at  somewhat  the  same  view  of  the 
nature  of  the  zone  of  weakness.  The  part  which  recrystallization  may  play  in  pro- 
moting such  movement  is  suggested  in  his  Research  in  China,  Vol.  II  (1907),  "Syste- 
matic Geology,"  pp.  130,  131. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  439 

pyrosphere  has  its  roots  in  the  asthenosphere.  But  in  the  attempt 
to  frame  a  logical  picture  of  the  processes  which  determine  the 
ascent  of  magmas,  the  question  arises  how  diffused  liquid  matter  is 
drained  away,  rising  and  uniting  at  higher  levels  into  magma 
reservoirs,  temporarily  molten,  and  the  direct  source  of  the  igneous 
activity  exhibited  within  the  outermost  crust.  A  deductive 
picture  is  as  follows — one  whose  truth  cannot  be  tested  directly, 
but  only  by  its  general  agreement  with  our  understanding. 

At  the  place  of  origin,  liquid  of  an  andesitic  or  basaltic  nature 
will  come  to  honeycomb  the  rock.  The  content  of  gases  is  pre- 
sumably sufficiently  high  to  reduce  the  viscosity.  The  liquid 
will  then  become  able  to  transmit  hydrostatic  pressures,  and, 
although  comprising  only  a  part  of  the  rock  mass,  will  constitute 
a  continuous  column  of  considerable  height.  Then  becomes 
possible  the  second  stage,  the  draining  upward  and  the  convergence 
of  the  fluid  rock.  Gravity  is  the  ultimate  cause,  as  in  the  down- 
ward movement  of  waters,  but  here  the  fluid,  being  lighter  than  its 
surroundings,  tends  to  move  upward.  An  explanation  of  this 
draining  process  has  been  given  by  Lane.1  In  a  gas-saturated 
rock  an  excess  of  gas,  or  liquid  and  gas,  would  have  the  power  of 
opening  fissures  at  any  depth  in  the  zone  of  flow  without  the 
necessity  for  the  existence  of  any  tensile  stress  in  the  walls,  or 
competence  in  the  walls  to  maintain  an  open  cavity.  All  that  is 
necessary  is  that  the  excess  pressure  in  the  rising  wedge  of  gas 
should  be  stronger  than  the  cohesion  of  the  rock.  The  fissure 
becomes  filled  with  gas  and  fluid  of  lesser  density  than  the  solid 
rock  of  the  walls.  Consequently,  the  pressure  transmitted  from 
below  is  greater  than  the  resisting  pressure  in  the  walls.  This 
insinuating  power,  owing  to  the  hydrostatic  head  due  to  the  lesser 
gravity  of  the  wedge,  becomes  greater  the  higher  the  wedge  rises 
above  the  source,  until  near  the  surface  the  action  may  become 
violent  and  rapid.  Daly  also  has  outlined  a  theory  of  mechanism 
for  the  injection  of  abyssal  wedges  of  magma  into  the  upper  crust.2 

'"Geologic  Activity  of  the  Earth's  Originally  Absorbed  Gases,"  Geological 
Society  of  America  Bull.,  V  (1894),  259-80. 

2  Am.  Jour.  Sci.,  XXII  (1906),  195-216;  Igneous  Rocks  and  Their  Origin  (1914), 
chap.  ix. 


440  JOSEPH  BARRELL 

His  theory  is  constructed  however  for  action  within  the  roof  of  a 
magmatic  substratum.  Daly  postulates  a  zone  of  tension,  but  the 
mechanism  suggested  by  Lane  does  not  require  this  and  would  seem 
to  apply  better  to  the  region  of  generation  of  magmas,  for  there  the 
cubic  compression  is  enormous,  it  is  not  a  zone  which  has  been 
subjected  to  cooling,  and  therefore  it  is  difficult  to  conceive  of  the 
cause  of  a  system  of  tensile  stresses  within  the  asthenosphere. 

We  are  now  prepared  to  draw  a  closer  analogy  between  the 
physical  conditions  of  the  asthenosphere  and  those  of  a  glacier, 
noting  the  likenesses  and  also  the  unlikenesses.  In  the  summer, 
in  the  case  of  Alpine  glaciers,  heat  is  supplied  to  the  surface  of  the 
glacier  until  it  is  warmed  to  the  melting-point,  and  part  of  the  ice 
absorbs  the  amount  required  by  the  change  of  state  and  passes  into 
water.  This  trickles  along  the  surface  until  a  fissure  is  met  and 
the  water  sinks  by  force  of  gravity  toward  the  bottom.  Near 
the  snout  of  the  glacier  the  temperature  of  this  deeper  part  may 
thus  be  more  dependent  upon  this  convection  than  upon  direct 
conduction  through  the  ice.  The  winter  freezing  tends  to  chill 
the  deeper  ice  and  slow  its  motion,  but  during  the  summer  the 
descending  water  tends  to  raise  the  temperature  toward  the  freezing- 
point.  In  parts  of  the  glacier  where  the  heating  is  more  effective 
than  the  cooling,  the  waters  drill  channels  and  gather  at  the  base 
of  the  glacier  into  streams,  reaching  finally  the  outer  world.  The 
descent,  and  gathering,  and  englacial  flow  of  glacial  waters  is 
analogous  to  the  rise  and  convergence  of  streamlets  of  molten  rock. 

In  order  to  account  for  lateral  mass  movement  within  the 
asthenosphere,  an  imperfection  of  isostasy  between  continental 
interior  and  ocean  basin,  giving  an  isostatic  gradient  or  slope  toward 
the  continental  interior,  seems  a  necessary  postulate.  Such  a 
gradient  is  so  low  that  it  has  not  yet  been  sifted  from  those  irregu- 
larities of  mass  which  are  owing  to  the  strength  of  the  outer  crust, 
or,  it  may  be,  obscured  by  great  compressive  bowings  of  the  crust. 
This  isostatic  gradient,  the  slope  required  to  generate  movement 
within  the  asthenosphere,  is  far  lower  than  that  of  the  surface  of  a 
continental  ice  sheet.  The  failures  of  the  analogy  between  the 
asthenospheric  and  glacial  states  are  then  as  instructive  as  the 
agreements.  The  glacier  is  thin  and  broad.  Friction  on  the  bot- 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  441 

torn  is  excessive  and  the  motion  requires  more  internal  work.  Much 
of  its  mass  is  permanently  well  below  the  freezing  temperature. 
These  are  the  factors  which  determine  the  steepness  of  the  surface 
gradient.  The  asthenosphere  by  contrast  should  be  deep  and  the 
differential  motions  within  it  necessary  to  satisfy  isostasy  would  be 
correspondingly  small.  The  temperature  through  a  wide  zone 
should  be  that  of  fusion  under  the  hydrostatic  pressures  prevailing. 
The  solid  rock  should  be  sodden  with  occluded  gases,  giving  mobility 
to  the  growing  fluid  and  ready  to  play  their  part  in  assisting  recrys- 
tallization.  Such  a  physical  condition,  as  long  as  there  is  a  continu- 
ous solid,  would  exhibit  perfect  elasticity  and  high  rigidity  during 
the  passage  of  transverse  vibrations,  yet  would  slowly  yield  to  pro- 
longed shearing  stresses,  even  though  these  were  very  small  in 
amount. 

RELATIONS    OF   IGNEOUS   ACTIVITY   TO   ASTHENOSPHERE   AND 
LITHOSPHERE 

The  argument  has  led  to  the  view  that  the  asthenosphere  is  a 
region  where  the  temperature  curve  becomes  tangent  to  the  fusion 
curve,  but  that  a  condition  of  solidity  is  maintained  by  the  recurrent 
elimination  of  that  material  which  becomes  molten.  The  impor- 
tance of  such  a  process,  maintaining  the  solidity  of  the  earth,  has 
been  dwelt  upon  by  Chamberlin,  especially  as  accounting  for  the 
overwhelming  igneous  activity  of  Archeozoic  time.  In  lessened 
measure  it  applies  to  all  later  times  as  well. 

Becker  has  held  that  the  bottom  of  the  zone  of  isostatic  com- 
pensation is  the  depth  at  which  the  temperature  curve  approached 
nearest  to  the  fusion  curve,  and  he  was  the  first  to  connect  in  this 
way  the  geodetic  evidence  with  a  temperature  relation.1  But 
Becker  does  not  conceive  of  actual,  permanent  contact  of  the  two 
curves  as  occurring,  and  took  the  depth  of  nearest  approach  to 
fusion  as  122  km.  This  follows  from  Hayford's  hypothesis  of  a 
uniform  distribution  for  isostatic  compensation,  but  in  the  present 
work  there  has  been  found  reason  for  believing  that  compensation 
fades  out  through  a  greater  depth;  the  strength,  as  measured  by 

1  "Age  of  a  Cooling  Globe  in  Which  the  Initial  Temperature  Increases  Directly 
as  the  Distance  from  the  Surface,"  Science,  XXVII  (1908),  227-33,  392>  "The  Age 
of  the  Earth,"  Smithsonian  Miscellaneous  Collections,  LVI,  No.  6  (1910),  1-28. 


442  JOSEPH  BARRELL 

the  existence  of  stress-difference,  through  a  depth  greater  still. 
The  beginning  of  permanent  contact  of  the  two  curves,  if  this  is 
the  cause  of  the  disappearance  of  strength,  should  be  as  much  as 
300  km.  deep  and  extend  through  some  hundreds  of  kilometers. 

A  rectilinear  projection  downward  of  the  temperature  gradient 
observed  at  the  surface  would  reach  the  fusion  temperature  of  rocks 
at  a  depth  of  about  50  km.  There  must  be  consequently  a  marked 
curvature  of  the  temperature  gradient  if  the  temperature  and 
fusion  curves  do  not  meet  short  of  300  km.  This  curvature  implies 
that  near  the  surface  there  is  either  a  greater  quantity  of  heat  flow- 
ing outward  by  conduction  or  that  the  conductivity  of  rock  near  the 
surface  is  very  greatly  decreased.  But  such  a  very  great  decrease 
in  conductivity  making  for  a  higher  temperature  gradient  finds  no 
supporting  evidence.  On  the  other  hand,  a  greater  outward  flow 
by  conduction  of  heat  near  the  surface  may  be  due  to  the  con- 
tinued generation  of  heat  by  radioactivity  to  a  greater  degree  than 
below;  or  also  to  a  rise  of  magmas  from  the  asthenosphere.  Mag- 
mas which  never  reach  the  surface  would  bring  heat  by  a  convective 
process  directly  into  the  outer  crust.  From  there  the  heat,  slowly 
diffused  upward  by  conduction,  would  increase  the  temperature 
gradient  in  the  outermost  part  of  the  lithosphere.  It  is  this  factor 
especially  which  the  argument  of  the  present  chapter  emphasizes. 

It  is  only  within  the  present  generation  that  general  recognition 
has  been  given  to  the  intrusive  nature  of  the  abyssal  igneous  rocks. 
They  are  now  generally  regarded  as  risen  from  the  depths.  Their 
action  has  been  to  break  through  and  engulf  the  foundations  of  the 
ancient  crust. 

This  process  of  batholithic  invasion  seems  to  be  recurrent  and 
widespread,  though  rise  into  the  outer  crust  is  restricted  to  the 
crises  of  diastrophism  and  usually  reaches  levels  exposed  to  erosion 
only  along  the  lines  of  mountain  systems.  The  stores*  of  heat 
brought  up  from  the  greater  depths  would  be  held  in  the  crust, 
especially  in  its  deeper  parts,  for  geologic  ages,  blurring  out  in  the 
course  of  time  by  conduction  and  creating  a  false  appearance  of 
heat  lingering  from  an  initial  molten  state,  a  resemblance  increased 
by  the  added  veil  of  new  heat  of  radio-active  origin  mantling  the 
ancient  stores. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  443 

The  temperature  gradient  under  this  view  should  naturally  vary 
widely  from  place  to  place  and  from  time  to  time.  Igneous  activity 
is  the  effective  means  by  which  heat  is  brought  up  from  depths 
which  on  account  of  the  slowness  of  conduction  would  be  otherwise 
thermally  isolated  from  the  outer  crust.  Offset  against  this,  cool- 
ing by  conduction  advances  downward  from  the  surface,  dissipating 
not  only  the  heat  of  local  radio-active  origin  but  that  excess  rising 
from  the  depths.  The  heat  of  the  crust  is  not  then  a  continually 
ebbing  residuum  from  a  primal  molten  state,  but  represents  rather 
an  oscillating  ebb  and  flow,  one  of  the  balances  of  nature  main- 
tained through  geologic  time. 

If  this  view  be  true — that  the  invasive  igneous  rocks  have  been 
an  important  factor  in  determining  the  amount  and  distribution  of 
heat  in  the  crust — it  is  doubtful  if  any  sound  arguments  can  be 
derived  from  the  study  of  the  present  gradients  as  to  the  initial 
temperature.  This  conclusion  is  similar  to  the  change  of  viewpoint 
in  other  lines  of  geology.  It  was  once  thought  that  the  composition 
of  the  present  atmosphere  and  the  character  of  present  climates  were 
steps  in  a  simple  and  continuous  series  of  changes  passing  from 
primal  conditions  to  a  future  in  which  the  water  would  be  absorbed 
into  the  earth  and  its  surface  transformed  into  a  frozen  desert. 
Now,  however,  it  is  generally  recognized  that  since  the  earliest 
known  times  the  surface  conditions  have  been  in  a  state  of  oscillat- 
ing equilibrium.  The  argument  of  this  section  leads  toward  the 
view  that  this  is  true  for  the  physical  conditions  within  lithosphere 
and  asthenosphere  also. 

[To  be  concluded] 


THE  STRENGTH  OF  THE  EARTH'S  CRUST— Concluded 


JOSEPH  BARRELL 

New  Haven,  Connecticut 


PART  VIII.     PHYSICAL  CONDITIONS  CONTROLLING  THE 
NATURE  OF  LITHOSPHERE  AND  ASTHENOSPHERE 

SECTION  B 
RELATIONS  WITH  OTHER  FIELDS  OF  GEOPHYSICS 

ERRONEOUS  CONCLUSIONS  REACHED  BY  THE  RECTILINEAR  PROJECTION 

or  SURFACE  CONDITIONS 499 

THE  EVIDENCE  OF  TIDES  ON  RIGIDITY  AND  STRENGTH  .  .  .  .504 
THE  EVIDENCE  OF  EARTHQUAKE  WAVES  ON  RIGIDITY  AND  DENSITY  .  506 
HIGH,  BUT  VARIABLE,  ELASTIC  LIMIT  WITHIN  THE  UPPER  LITHOSPHERE  509 
MODES  OF  LITHOSPHERIC  YIELDING  AND  THEIR  RELATION  TO  STRENGTH  512 

RELATIONS  WITH  OTHER  FIELDS  OF  GEOPHYSICS 

ERRONEOUS  CONCLUSIONS  REACHED  BY  THE  RECTILINEAR  PROJECTION 
OF  SURFACE  CONDITIONS 

Geologists  early  became  aware  that  temperature  increased  with 
depth.  Projecting  this  gradient  as  a  straight  line  indicated  that 
at  no  great  depth  the  temperature  was  sufficiently  high  to  melt  all 
rocks  and,  in  testimony,  volcanoes  brought  such  melted  rocks  to 
the  surface.  The  earth  was  consequently  looked  upon  as  a  molten 
or  even  gaseous  body  enveloped  by  a  thin  crust  of  solid  rock.  The 
logic  of  this  conclusion  seemed  incontrovertible  and  moreover  it 
was  in  accord  with  the  simpler  expectations  from  the  nebular 
hypothesis.  Nevertheless,  direct  and  positive  evidence  from  sev- 
eral independent  sources  has  forced  on  geologists  the  belief  that 
the  earth  is  not  only  solid  throughout,  but,  as  a  whole,  is  more 
rigid  than  steel.  Slowly  and  with  difficulty  the  older  view  has 
therefore  had  to  be  abandoned.  Yet  it  continually  recurs  in  one 
form  or  another,  advocated  chiefly  by  writers  who  see  the  direction 
in  which  the  surface  evidence  of  temperature  gradient  leads,  who 
regard  it  as  compulsory,  and  who  do  not  recognize  or  give  equal 

.    499 


500  JOSEPH  BARRELL 

weight  to  the  direct  evidence  regarding  the  nature  of  the  earth's 
interior.  Because  of  the  ease  and  certainty  of  laboratory  studies 
there  is  a  tendency  to  treat  the  interior  of  the  earth  as  though 
it  were  incapable  of  speaking  for  itself  through  the  evidence  of 
geophysics,  geodesy,  and  geology,  but  must  remain  forever  a  play- 
ground for  the  speculative  imagination.  Largely  unknown  the 
nature  of  the  earth's  interior  is  and  long  must  be;  laboratory 
studies  on  the  influences  of  heat,  of  pressure,  and  of  chemical 
composition,  upon  the  physical  state  of  the  crust,  must  constitute 
the  paths  which  guide  in  the  search  downward  into  the  unknown; 
but  the  final  test  of  hypothesis  must  be  the  direct  testimony  of  the 
earth  itself. 

The  rectilinear  projection  of  surface  conditions  is  based  on  the 
assumption  that  the  temperature  gradient  is  a  straight  line  to 
great  depths,  or  that  strength,  or  density,  or  porosity,  as  the  case 
may  be,  is  not  changed  by  the  pressures  of  the  interior.  Such 
assumptions  lead  to  views  more  or  less  in  opposition  to  those 
reached  in  the  present  investigation.  They  must,  therefore,  be 
discussed  to  some  degree.  An  illustration  of  these  dangers  of 
reasoning  by  unchecked  extrapolation  is  supplied  by  a  paper 
written  by  Arrhenius,1  selected  for  discussion  because  of  the  emi- 
nence of  the  author  in  the  fields  of  physics  and  chemistry,  the 
definiteness  with  which  his  conclusions  are  stated,  and  the  wide 
citation  which  this  paper  has  achieved.2  In  this  paper  the  argu- 
ments are  given  in  favor  of  a  gaseous  nature  of  the  interior  of  the 
earth,  carrying  forward  an  idea  first  suggested  by  A.  Ritter  in  a 
series  of  "Researches  on  the  Height  of  the  Atmosphere  and  the 
Constitution  of  Gaseous  Heavenly  Bodies." 

From  the  rate  of  increase  of  temperature  with  depth  Arrhenius 
argues  that  at  a  depth  of  40  km.  the  crust  must  pass  into  a  molten 
condition,  but  one  which,  because  of  pressure,  is  a  viscous  and 
highly  incompressible  liquid.  At  a  depth  of  some  300  km.  the 
temperature,  he  states,  must  be  above  the  critical  temperature  of 

1  "Zur  Physik  des  Vulkanismus,"  Geol.  Foren.  i  Stockholm,  Forhandl.,  XXII 
(1900),  395-4I9- 

2  See  for  example  its  presentation  by  A.  Geikie,  Text  Book  of  Geology,  Vol.  I  (1903), 
pp.  71-74. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  501 

all  known  substances,  and  therefore  the  liquid  magma  passes  into 
a  gaseous  magma  extending  to  the  center  of  the  earth.  The  author 
then  notes  that  the  chemical  elements  of  highest  atomic  weight 
are  not  detected  in  the  sun,  but  states  that  without  doubt  they 
occur,  and  concludes  from  this  that  they  must  be  concentrated 
by  virtue  of  gravity  toward  the  sun's  center.1  The  high  density 
of  the  earth's  interior  is  accordingly  to  be  explained  by  the  presence 
of  substances  heavier  than  surface  rocks.  For  many  reasons,  as 
the  dominance  of  iron  in  nature,  as  shown  by  meteorites,  by  the 
spectrum  of  the  sun,  and  by  the  magnetism  of  the  earth,  it  is  to  be 
concluded  that  this  substance  which  he  thinks  necessary  to  account 
for  the  high  density  of  the  earth's  interior  is  metallic  iron.  The 
earth  consists  consequently  of  the  following  portions  measured 
from  the  center  on  the  radius.  Eighty  per  cent  of  the  radius  is 
gaseous  iron,  15  per  cent  is  gaseous  rock  magma,  about  4  per  cent 
is  fluid  rock  magma,  and  somewhat  less  than  one  per  cent  is  solid 
crust.2 

To  reconcile  these  conclusions  with  the  incontrovertible  evi- 
dence of  rigidity,  Arrhenius  takes  up  another  line  of  rectilinear 
extrapolation  and  carries  it  to  an  equally  extreme  degree.  Fluids 
in  general  show  a  somewhat  readier  compressibility  than  solids. 
At  high  pressures  then  it  is  argued  that  liquids  will  customarily 
occupy  less  volume  than  solids  and  the  pressure  will  tend  to  lower, 
not  raise,  the  melting-point.  Consequently;  the  rigidity  cannot 
be  accounted  for  by  the  maintenance  of  solidity  through  pressure. 
The  author  then  points  out  that  under  enormous  pressures  all  sub- 
stances, even  gases,  must  become  highly  incompressible;  and  that 
at  high  temperatures,  where  the  volume  is  maintained  the  same,  the 
viscosity  of  gases  or  fluids  increases  with  increase  in  temperature. 
From  this  it  is  argued  that  in  the  central  parts  of  the  earth  gaseous 
iron  is  more  incompressible  and  viscous  than  solid  steel.  It  is 
by  enormous  pressure  consequently  in  spite  of  a  gaseous  nature 
that  the  interior  of  the  earth  exhibits  its  great  rigidity. 

Vulcanism  according  to  Arrhenius  is  connected  with  the  free 
seepage  of  ocean  water  downward  through  the  crust  which,  he 
holds,  constitutes  a  semipermeable  membrane.  By  the  absorp- 

"  Op.  cit.,  p.  402.  2  Op.  cit.,  p.  405. 


502  JOSEPH  BARRELL 

tion  of  water  into  the  heated  rocks  the  conditions  for  volcanic 
activity  are  initiated.  This  argument,  like  the  others,  is  in  the 
form  of  a  great  extension  or  extrapolation  of  factors  operative  in 
a  small  way  in  the  laboratory  to  conditions  in  nature  which  are 
wholly  different  in  magnitude. 

As  comments  upon  this  paper,  it  should  be  noted  that  nearly 
every  conclusion  applying  to  the  sun  and  earth  may  be  questioned. 

At  a  depth  of  i  ,000  km.,  according  to  Arrhenius,  the  temperature 
is  about  30,000°  C.  The  gradient  is  thus  taken  as  essentially  a 
straight  line  from  the  surface  downward.  There  is  no  demon- 
stration as  to  why  this  rectilinear  extension  is  assumed,  whether 
it  is  to  be  regarded  as  an  adiabatic  temperature  curve  produced  by 
condensation  under  pressure  or  produced  in  some  other  way.  The 
influence  of  cooling  through  geologic  time  in  changing  the  outer 
gradient  is  not  considered;  nor  the  influence  of  rising  magmas. 
The  existence  of  radioactivity  was  then  just  beginning  to  be  appre- 
ciated and  naturally  could  not  have  been  evaluated,  but  the  data 
for  a  discussion  of  the  other  factors,  though  at  hand,  was  neglected. 

There  is  no  demonstration  that  the  heavy  elements  are  con- 
centrated in  the  sun's  interior,  or  that  the  earth  is  mostly  metallic 
iron.  It  is  possible  that  the  earth  is  thus  constituted,  but  it  must 
be  proved  on  better  evidence  than  a  citation  of  the  dominance  of 
iron  in  nature.  The  incompressibility  of  all  substances,  both 
fluid  and  liquid,  increases  greatly  with  great  increase  of  pressure, 
following  apparently  parabolic  curves.  Therefore,  it  cannot  be 
argued  with  any  assurance  that  the  high  incompressibility  of  the 
earth's  interior  proves  the  presence  of  iron,  or  that  under  such 
pressures  the  fluid  occupies  less  volume  than  the  solid  state. 

At  a  depth  of  1,000  km.  Arrhenius  states  that  the  temperature 
is  about  30,000°  C.  and  the  pressure  250,000  atmospheres.1  If, 
under  these  conditions  of  exalted  temperatures,  gaseous  rock  or 
iron  has  a  viscosity  equal  to  that  of  solid  steel  it  may  well  be  asked 
how  the  stars,  with  their  immensely  greater  masses  and  consequent 
internal  pressures,  can  maintain  a  convective  circulation  compe- 
tent to  keep  up  their  enormous  surfa.ce  radiation.  Furthermore, 
however  viscous  a  compressed  gas  or  liquid  may  be,  this  property 

1  Op.  cit.,  p.  400. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  503 

should  be  distinguished  from  rigidity.  If  a  body  can  resist  even 
small  shearing  stresses  for  an  indefinite  period,  it  has  the  essential 
properties  of  a  solid  and  not  a  gas.  If  it  possesses  real  rigidity, 
even  if  it  should  be  true  that  under  relief  from  pressure  the  sub- 
stance would  turn  into  a  gas,  yet  such  relief  cannot  take  place  and 
it  is  a  confusion  of  terms  to  speak  of  the  substance  as  a  gas  when 
exhibiting  to  a  striking  degree  the  essential  qualities  of  a  solid. 
This  distinction  between  viscosity  and  rigidity  is  of  first  importance, 
yet  is  not  mentioned  by  Arrhenius.  Although  undercooling  of 
a  fluid  into  a  glass  gives  rise  to  the  elastic  properties  of  a  solid, 
it  has  not  been  shown  that  increase  of  pressure,  however  great, 
upon  a  gas  above  the  critical  temperature  would  transform  in- 
creasing fluid  viscosity  into  solid  rigidity  and  plasticity  such  as  is 
exhibited  by  the  earth. 

As  to  the  hypothesis  that  the  crust  is  a  semipermeable  mem- 
brane, permitting  a  free  downward  seepage  of  ocean  water,  but 
little  need  be  said,  since  this  is  a  subject  which  has  been  much 
discussed  in  recent  years  and  is  now  largely  discarded  by  geologists. 
The  evidence  against  it  is  varied.  Petro logic  study  shows  the 
deep  rocks  to  be  impermeable  and  unaltered;  beyond  a  shallow 
depth  they  are  dry,  and  their  gaseous  and  liquid  occlusions  are 
held  unchanged  for  geologic  ages.  Unsound  conclusions  have  been 
built  upon  the  behavior  of  steam  within  porous  sandstones,  com- 
bined with  confusion  of  the  rate  of  diffusion  under  enormous 
pressure-gradients  in  the  laboratory  with  enormous  pressures,  but 
low  pressure-gradients  within  the  crust.  Furthermore  volcanoes 
are  not  restricted  to  the  vicinity  of  the  sea  and  their  emanations 
are  not  of  the  proper  composition  to  have  been  derived  from  ocean 
waters.  As  Suess  has  said,  volcanoes  are  not  nourished  by  the 
sea,  but  every  volcanic  eruption  adds  to  the  waters  of  the  ocean. 

The  paper  under  discussion  was  written  by  a  scientist  who  has 
done  much  exact  work  in  physical  chemistry,  but  who  in  passing 
to  geologic  thinking  has  adopted  the  habit  of  an  earlier  generation 
— a  habit  of  speculative  thought,  suggested  by  chemical  and  physi- 
cal concepts  and  not  verified  by  a  study  of  the  earth.  The  form 
of  present  geologic  investigations  has,  however,  advanced  to  the 
quantitative  stage,  although  the  data  are  often  so  inexact  that  the 


504  JOSEPH  BARRELL 

order  of  magnitude,  or  the  direction  of  the  truth,  is  all  which  may 
be  now  ascertainable. 

In  conclusion,  it  is  seen  that  the  hypotheses  outlined  by  Arrhe- 
nius  imply  a  thinness  of  the  crystalline  lithosphere  and  a  crustal 
weakness  wholly  at  variance  with  the  conclusions  regarding  strength 
which  have  been  reached  in  this  investigation.  They  imply  a 
difference  in  nature  of  the  earth's  interior  from  that  given  by  the 
more  direct  lines  of  evidence,  as  shown  by  the  body  resistance  of 
the  earth  to  vibratory  distortions  of  both  short  and  long  periods. 
Because  of  these  many  difficulties,  this  group  of  hypotheses, 
adopted  by  Arrhenius,  has  already  been  largely  discarded,  though 
they  still  find  considerable  acceptance,  more  especially  by  workers 
in  related  fields  of  science.  But  the  measures  of  lithospheric 
depth  and  strength  which  appear  to  be  given  by  geodesy  add  their 
testimony  to  the  cumulative  evidence  against  these  views. 

THE   EVIDENCE    OF   TIDES    ON   RIGIDITY   AND   STRENGTH 

The  tidal  distortion  of  the  solid  earth  measured  by  means  of 
the  horizontal  pendulum  has  shown  that  its  rigidity  is  of  the  order 
of  magnitude  of  steel.  But  the  recent  measurements  by  Michelson 
and  others,  employing  a  long  horizontal  pipe  partly  filled  with 
water,  showed  clearly  that  the  earth's  rigidity  is  even  greater  than 
that  of  steel.1  This  higher  value  is  in  agreement  with  the  induc- 
tions from  the  observations  on  the  variations  of  latitude.  But 
these  measurements  give  the  rigidity  of  the  earth  as  a  whole,  not 
the  distribution  of  rigidity.  The  resistance  to  tidal  deformation 
is  furthermore  complicated  by  the  influence  of  gravity  and  increas- 
ing density  with  greater  depth.  Even  if  the  earth  were  a  liquid 
globe  it  would  resist  tidal  distortion  to  one-third  the  degree  of  the 
resistance  of  a  globe  of  steel,  and  if  the  liquid  sphere  were  denser 
inside,  this  ratio  would  be  further  decreased.2  Notwithstanding 
this  factor,  however,  it  is  clear  that  the  earth  as  a  whole  is  more 
rigid  than  steel.  As  the  outer  part  is  known  to  be  less  rigid  than 
steel,  it  follows  that  the  rigidity  of  much  of  the  interior  must  be 

1  "Preliminary  Results  of  Measurements  of  the  Rigidity  of  the  Earth,"  Jour. 
Go?/.,  XXII  (1914),  118. 

2  A.  E.  H.  Love,  Elasticity,  p.  306. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  505 

proportionately  higher.  But  the  tidal  stresses,  though  serving 
as  a  measure  of  the  rigidity  of  the  earth  as  a  whole,  are  so  small 
that  they  are  ineffective  as  a  measure  of  the  strength  of  the  earth 
as  a  whole,  or  of  even  its  weakest  parts.  The  smallness  of  the 
stresses  can  be  appreciated  by  noting  Darwin's  numerical  calcu- 
lations. In  his  original  paper  Darwin  arrived  at  the  conclusion 
that  the  tidal  stress-differences  at  the  center  of  the  earth  were 
eight  times  as  great  as  at  the  surface,  and  this  result  has  been 
widely  quoted.  In  the  final  publication,  however,  a  correction 
•is  made  showing  that  this  is  the  ratio  between  the  surface  stress 
at  the  poles  as  compared  to  the  center.  The  stresses  at  the  poles, 
at  the  equator,  and  at  the  center  he  finds  to  be  in  the  ratio  of  i  to  3 
to  8.  The  diurnal  tide  gives  an  actual  stress-difference  per  square 
centimeter  amounting  to  16  grams  at  the  poles,  48  at  the  equator, 
and  128  at  the  earth's  center.1  The  strength  of  granite  at  the 
surface  of  the  earth  averages  about  1,700,000-2,000,000  grams  per 
square  centimeter.  The  elastic  limit  for  steel  subjected  to  tensile 
or  compressive  stresses  in  one  direction  ranges  from  about  3,500,000 
grams  to  4,500,000  grams  per  square  centimeter,  according  to  the 
grade  of  the  metal.  The  ultimate  strength  is  about  twice  as  high 
as  the  elastic  limit.  Thus  the  earth  is  stressed  by  the  tidal  forces 
even  at  the  center  to  only  about  one  part  in  fifteen  thousand  of  the 
strength  of  good  granite  at  the  surface,  or  about  one  part  in  twenty- 
seven  to  thirty-five  thousand  of  the  limits  of  perfect  elasticity 
which  steel  exhibits  in  the  laboratory.  With  stresses  so  small 
it  is  not  surprising  that  although  tides  give  measurements  of 
rigidity  their  evidence  regarding  viscosity  is  most  uncertain.  The 
results  of  estimates  of  the  viscosity  are  more  or  less  contradictory 
and  so  small  as  to  be  within  the  probable  error  of  determination. 
Nevertheless  Schweydar  considers  that  there  is  a  suggestion  of  a 
slightly  plastic  zone  extending  from  a  depth  of  about  120  to  620  km. 
Although  this  has  been  adopted  in  the  present  article  as  the  limit 
of  the  asthenosphere,  it  would  appear  that  the  convincing  proof  for 
the  existence  of  such  a  zone,  and  the  determination  of  its  limits 

1  George  H.  Darwin,  "On  the  Stresses  Caused  in  the  Interior  of  the  Earth  by 
the  Weight  of  Continents  and  Mountains,"  Collected  Scientific  Papers  (1908),  II, 
p.  481;  original  publications,  Phil.  Trans.  Roy.  Soc.,  CLXXIII  (1882),  187-223,  and 
Proc.  Roy.  Soc.,  XXXVIII  (1885),  322-28. 


506  JOSEPH  BARRELL 

also  is  more  likely  to  be  given  by  the  geologic  and  geodetic  evidence 
rather  than  from  that  yielded  by  the  tides,  provided  that  the  present 
hypothesis  of  the  existence  of  an  asthenosphere  is  accepted. 

It  might  seem  that  if  the  asthenosphere  is  strained  to  its  limit 
by  permanent  stress  and  is  slowly  yielding,  that  even  the  small 
and  rhythmic  tidal  stresses,  like  the  last  straw  on  the  camePs 
back,  might  reveal  a  lack  of  resilience  in  the  region  of  yielding. 
The  distinction  was  emphasized  in  Section  A,  however,  that  an 
elastic  limit  which  is  determined  for  permanent  stress  by  a  facility 
of  recrystallization  at  a  high  temperature  may  be  a  far  lower  elastic 
limit  than  that  which  would  exist  for  rapid  rhythmic  stresses. 
Recrystallization  would  theoretically  go  forward  a  little  more 
rapidly  during  the  additive  phase  of  the  tidal  stress,  but  the  process 
is  presumably  so  slow,  and  the  tidal  stress  so  small  and  rapid,  that 
no  appreciable  effects  would  be  attained  before  the  following  of  the 
negative  phase.  A  high  resilience  of  the  earth  under  tidal  stress 
seems  therefore  quite  compatible  with  the  existence  of  a  slowly 
yielding  asthenosphere. 

THE   EVIDENCE   OF   EARTHQUAKE   WAVE§   ON  RIGIDITY  AND  DENSITY 

The  speed  of  an  elastic  wave  through  a  solid  varies  directly 
with  the  square  root  of  the  modulus  of  elasticity  and  inversely 
with  the  square  root  of  the  density.  There  are  two  waves,  corre- 
sponding to  the  elasticities  of  volume  and  form  respectively,  the 
one  measured  by  the  modulus  of  compressibility,  the  other  by  the 
modulus  of  rigidity.  The  first  is  the  longitudinal  or  radial  wave, 
the  second  is  the  transverse  wave.  The  former  outruns  the  latter 
and  gives  rise  to  the  first  preliminary  tremor  by  which  the  earth- 
quake records  itself  in  distant  regions.  The  transverse  vibration 
is  felt  as  the  second  preliminary  tremor,  followed  by  the  much 
larger  oscillations  of  the  principal  wave.  The  first  two  go  through 
the  earth,  the  latter  passes  around  the  surface.  The  fact  that 
there  is  a  transverse  wave  shows  that  the  earth  is  solid  throughout. 
But  the  vibrations  at  the  point  of  emergence  for  waves  which  have 
penetrated  more  than  half-way  into  the  earth  are  so  faint  because 
of  distance  that  their  beginnings  are  in  doubt,  and  consequently 
the  speeds  of  transmission  below  one-half  of  the  radius  are  uncer- 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  507 

tain.  These  greater  depths  do  not,  however,  so  immediately 
concern  the  present  subject.  For  the  outer  quarter  of  the  earth 
both  radial  and  transverse  waves  increase  in  velocity  of  trans- 
mission with  depth,  showing  that  incompressibility  and  rigidity 
increase  faster  than  density  and  reach  values  greater  than  those 
exhibited  by  steel  at  the  surface  of  the  earth.1 

So  much  is  certain,  but  when  it  comes  to  testing  the  character 
of  any  particular  shell  by  means  of  the  velocities  and  character  of 
the  vibrations  which  have  passed  through  it,  there  is  but  little 
certainty.  The  difficulty  of  an  exact  interpretation  is  discussed 
well  by  Knott.2  To  illustrate  the  variety  of  opinions,  Benndorf 
has  worked  out  a  law  according  to  which  the  speed  of  transmission 
increases  rapidly  to  a  depth  of  200  miles  (320  km.)  from  the  sur- 
face. Knott  assumes  a  constancy  of  speed  below  a  depth  of  400 
miles  (644km.).3  Wiechert  has  concluded  that  there  are  sudden 
changes  in  velocity  at  depths  of  1200, 1650,  and  2450  km.  Poisson's 
ratio  which  expresses  the  relationships  of  the  elasticities  of  form 
and  volume  remains,  however,  practically  constant  throughout, 
having  a  mean  value  of  o.27.4  These  changes  imply  surfaces  of 
discontinuity.  If  real,  however,  they  are  deeper  than  the  shells 
of  the  earth  involved  in  the  problems  of  isostasy.  The  conclusions 
rest,  however,  upon  data  of  doubtful  reliability.  Reid  has  made 
a  critical  examination  of  this  subject  in  connection  with  his  com- 
prehensive study  of  the  excellent  records  obtained  from  many 
parts  of  the  world  of  the  California  earthquake  of  igo6.s  Follow- 
ing Wiechert's  method,  the  curves  representing  the  normals  to 
the  wave  fronts  and  the  velocities  at  various  depths  were  computed 
from  the  data  of  the  seismograms.  The  result  showed  that  for  the 
radial  or  longitudinal  wave  the  velocity  increased  rapidly  with 
depth  but  with  decreasing  rapidity,  from  7 .  2  km.  per  second  at  the 

1  Galitzen,  Vorlesungen  iiber  Seismometrie,  p.  138,  1914. 

2  Physics  of  Earthquake  Phenomena  (1908),  chap.  xii. 
*Op.  cit.,  pp.  248-50. 

4G.W.  Walker,  Modern  Seismology,  1913,  p.  61. 

5  California  Earthquake  of  April  18,  1906  (Report  of  the  State  Earthquake  Inves- 
tigation Commission,  Vol.  II,  "The  Mechanics  of  the  Earthquake,"  by  H.  F.  Reid). 
Published  by  the  Carnegie  Institution  of  Washington,  1910. 


.  508  JOSEPH  BARRELL 

surface  to  12.5  km.  per  second  at  2,170  km.  from  the  surface,  o. 66 
of  the  radius  from  the  center.  Below  that  depth  the  velocity  is 
nearly  constant.  The  velocity  of  the  transverse  waves  is  4.8  km. 
per  second  at  the  surface  and  increases  almost  linearly  with  depth, 
reaching  a  velocity  of  about  7 . 5  km.  per  second  at  half  the  distance 
to  the  center  of  the  earth.  The  absence  of  good  records  from 
distances  beyond  125°  prevents  a  knowledge  of  the  velocities  at 
greater  depths.  Within  the  limits  regarding  which  information  is 
given,  Reid  remarks  that  there  is  no  indication  of  a  sudden  change 
in  the  velocity  of  either  wave  such  as  we  should  expect  if  there  were 
any  sudden  changes  in  the  nature  of  the  earth's  interior.  Oldham 
also  finds  no  evidence  of  sudden  change  to  a  depth  of  at  least 
2,400  miles,  0.4  radius  from  the  center.1  From  the  curves  showing 
the  relation  of  velocity  to  depth  which  Reid  gives2  it  is  seen  that 
the  ratio  of  velocity  of  the  transverse  to  the  velocity  of  the  longi- 
tudinal wave  is  o .  66  at  the  surface,  0.56  at  o .  95  R,  o .  53  at  o .  9  R, 
reaching  a  minimum  of  0.52  at  o. 85  R,  from  which  it  increases  to 
o.  58  at  o.  5  R.  This  shows  that  both  moduli  of  elasticity  increase 
with  depth,  but  that  down  to  a  depth  of  between  0.8  and  0.9  R. 
from  the  center  of  the  earth,  637  and  1,274  km.  from  the  surface,  in- 
compressibility  increases  relatively  faster  than  rigidity.  The  change 
is  shown  as  very  rapid  in  the  first  300  km.  This  is  the  only  way 
in  which  the  existence  of  an  asthenosphere  reflects  itself  in  the 
rigidity  of  the  earth,  and  this  may  not  be  related  to  its  weakness 
but  to  some  other  property,  such  as  the  nature  of  compressibility 
or  of  changing  chemical  composition,  or  partly  in  the  lack  of 
detailed  knowledge  in  the  nature  of  the  data. 

Earthquake  waves,  like  the  tides,  measure  elasticity  rather 
than  strength.  The  vibrations  which  penetrate  200-300  km., 
and  more,  downward  in  the  earth  are  already  greatly  reduced  in 
amplitude  and  therefore  in  the  strains  which  they  bring  on  the 
earth.  What  the  maximum  strains  may  be  is  unknown,  but  reason- 
able assumptions  as  to  amplitude  show  that  within  the  astheno- 
sphere the  order  of  magnitude  of  the  strains  would  be  of  the  nature 

1  "On  the  Constitution  of  the  Interior  of  the  Earth  as  Revealed  by  Earthquakes," 
Quar.  Jour.  Geol.  Soc.,  LXII  (1906),  p.  470. 

2  P.   122. 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  509 

of  a  thousandth  part  of  that  which  granite  at  the  surface  of  the 
earth  can  sustain.  Furthermore,  even  if  the  stresses  were  greater 
and  could  be  used  as  a  measure  of  strength,  this  would  apply  to 
sudden  stresses  only  and  the  results  obtained  from  elastic  vibra- 
tions could  not  be  used  safely  as  a  means  of  determining  the 
strength  under  long-enduring  stresses.  Thus  the  evidence  from 
both  tides  and  earthquakes  is  negative  in  regard  to  the  existence 
of  an  asthenosphere.  They  show  only  that  it  is  not  fluid  and  that 
it  is  not  markedly  unlike  the  rest  of  the  earth  in  its  elastic  properties. 

HIGH,  BUT  VARIABLE,  ELASTIC  LIMIT  WITHIN  THE  UPPER  LITHOSPHERE 

The  experiments  by  F.  D.  Adams  showed  that  under  conditions 
of  cubic  compression  rocks  became  far  stronger  than  when  sub- 
jected to  compression,  as  at  the  surface  of  the  earth,  in  one  direction 
only.  When  a  cylinder  of  Westerly  granite  was  incased  in  a  steel 
jacket  and  then  subjected  to  heavy  pressure  upon  its  ends,  a  small 
cavity  within  the  specimen  just  began  to  break  down  under  a  stress- 
difference  of  between  160,000  and  200,000  pounds  per  square  inch, 
about  six  to  eight  times  the  strength  possessed  by  this  rock  under 
surface  conditions.  At  a  temperature  of  550°  C.,  a  temperature 
calculated  to  exist  at  a  depth  of  n  miles  below  the  earth's  surface, 
small  cavities  remained  open  when  submitted  to  considerably 
greater  pressures  than  occur  from  the  overlying  load  at  this 
depth.1 

Adams'  experiments  and  King's  calculations  are  most  important 
and  show  without  doubt  that  the  more  superficial  parts  of  the 
earth,  to  a  depth  of  ten  to  fifteen  miles  at  least,  are  far  stronger 
than  had  been  supposed;  but  they  apply  to  the  temperature  and 
pressure  gradients  in  places  of  geologic  quiet,  not  to  regions  under- 
going igneous  intrusion  and  crustal  deformation.  Then  the  tem- 
peratures may  become  far  higher  and  the  crust  surcharged  with 
magmatic  gases.  Yet  it  is  under  these  conditions  especially,  of 
geologic  activity  as  contrasted  to  geologic  quiet,  that  regional  meta- 
morphism  and  rock  flowage  proceeds.  Still  less  does  this  experi- 
mental work  prove  a  great  strength  of  the  crust  at  depths  of  more 

1  Louis  Vessot  King,  "On  the  Limiting  Strength  of  Rocks  under  Conditions  of 
Stress  Existing  in  the  Earth's  Interior,"  Jour.  GeoL,  XX  (1912),  136,  137. 


510  JOSEPH  BARRELL 

than  a  hundred  kilometers,  for  there  the  temperatures  are  presum- 
ably above  those  which  under  the  conditions  of  freedom  from  pres- 
sure at  •  the  surface  of  the  earth  produce  dry  fusion.  Occluded 
gases,  furthermore,  are  held  beyond  possibility  of  escape. 

The  strength  of  the  crust  is  dependent  consequently  upon  four- 
fold conditions — the  nature  of  the  material,  the  cubic  compression, 
the  relation  of  temperature  to  the  point  of  fusion,  and  the  rapidity 
of  the  application  of  the  stress.  These  factors  are  all  variable  with 
time  and  place.  How  variable  will  be  seen  upon  further  consider- 
ation in  the  following  paragraphs. 

The  influence  of  the  nature  of  the  material  is  seen  when  it  is 
noted  that  granite  is  only  about  one-half  as  rigid  as  the  basic  rocks, 
although  it  is  not  less  strong.  Consequently,  regional  stress  coming 
upon  a  complex  of  two  such  rocks  will  elastically  deform  the  granite 
more  readily,  a  greater  stress  will  be  thrown  upon  the  basic  rocks, 
and  since  their  elastic  limit  is  not  correspondingly  higher  they 
should  begin  to  yield  by  flow  or  fracture  before  the  more  pliant 
rocks  had  reached  their  limit.  The  general  conclusion  is  that 
a  movement  of  compression  in  the  earth's  crust  must  necessarily 
give  rise  to  unequal  strains  and  concentration  of  stress,  as  well 
from  variations  in  chemical  composition  as  from  variations  in 
structure.  The  local  stress  may  rise  far  higher  than  the  general 
regional  stress. 

As  to  the  second  factor,  during  the  progress  of  normal  fault- 
ing the  horizontal  compressive  stress  in  the  crust  is  less  than  the 
vertical  stress  due  to  weight.  During  the  progress  of  folding  and 
mashing,  on  the  contrary,  the  horizontal  stresses  become  far 
higher.  But  the  least  of  the  three  principal  stresses  determines 
the  amount  of  cubic  compression;  the  difference  between  the 
greatest  and  least  stresses  determines,  on  the  contrary,  the  amount 
and  direction  of  the  strain  upon  the  rigidity  of  the  rock.  Thus  it  is 
seen  that  both  the  cubic  compression  and  the  stress-difference  vary 
with  the  amount  and  kind  of  forces. 

It  is  temperature,  however,  which  is  probably  the  most  variable 
of  these  factors.  Igneous  activity  brings  the  temperatures  of  the 
greater  depths  comparatively  near  to  the  surface  and  must  produce 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  511 

widespread  weakening  of  the  crust,  both  through  the  physico- 
chemical  effects  of  the  exalted  temperatures  and  the  structural 
effects  of  the  intruded  viscous  fluids. 

The  rapidity  of  the  application  of  stress  is  a  variable  in  itself 
and  furthermore  has  variable  effects,  but  would  seem,  however, 
to  be  the  least  important  of  these  several  factors.  The  movements 
of  horizontal  compression  and  vertical  warping  are  slow  and  give 
time  for  recrystallization  in  the  deeper  crust.  In  this  way  they 
meet  a  lesser  resistance  than  would  rapid  stresses.  Where  the 
temperatures  are  close  to  those  of  fusion  it  would  seem  in  fact  that 
rock  flowage  by  recrystallization,  developing  the  gneissoid  structure, 
should  demand  markedly  less  shearing  stress  than  the  process  of 
granulation.  The  gnarled  and  twisted  rocks  of  the  Archean  speak 
of  the  presence  beneath  them  of  molten  magmas  rather  than  of  an 
enormous  degree  of  compressive  forces  upon  them.  But  ready 
yielding  by  recrystallization  in  one  place  would  permit  the  con- 
centration of  mashing  stresses  upon  other  localities  and  raise  the 
strain  to  that  intensity  needed  for  granulation.  An  enormous 
depth  of  cover,  such  as  Adams'  experiments  have  been  thought  to 
show,  is  not  suggested  by  the  geologic  evidence,  nor  apparently 
is  it  demanded  by  a  completer  theory. 

In  fault  movements  and  in  dike  or  sheet  intrusion  accompanied 
by  the  expansion  of  gases  are  two  sources  of  rapid  application  of 
forces.  It  is  probable,  however,  that  their  deformative  action  is 
confined  to  the  outer  ten  miles  of  the  crust,  and  their  consider- 
ation need  not  detain  us  in  the  evaluation  of  those  factors  of 
strength  which  concern  the  crust  as  a  whole. 

Summing  up  the  conclusions  from  these  various  lines  of  evidence, 
physical  and  geological,  it  is  seen  that  they  suggest  a  rapid  increase 
of  strength  with  depth,  then  the  gradual  passage  into  a  deep  zone 
of  lowered  strength.  The  limits  and  values,  however,  are  variable 
with  time  and  place.  Such  a  distribution  of  strength  as  is  indi- 
cated by  these  independent  lines  is  in  accord  with  the  interpretation 
of  the  geodetic  evidence  showing  the  existence  of  crustal  compe- 
tence to  support  heavy  loads  over  certain  limits  of  area,  coexist- 
ing with  flotational  equilibrium  over  much  broader  regions. 


512  JOSEPH  BARRELL 

MODES   OF   LITHOSPHERIC   YIELDING  AND   THEIR  RELATION 
TO    STRENGTH 

The  relationship  of  strength  to  depth  which  has  been  derived 
in  this  study  and  which  was  expressed  in  the  curve  of  strength  at 
the  end  of  Part  VII  is  to  be  connected  with  the  physical  qualities 
discussed  in  this  part.  Here  it  is  seen  that  it  is  a  curve  of  elastic 
limit.  When  that  limit  is  exceeded,  permanent  deformation  must 
take  place;  by  one  means  at  the  surface,  by  another  within  the 
body  of  the  lithosphere,  by  still  another  at  its  base. 

At  the  surface  the  typical  mode  of  yielding  is  by  jointing  and 
faulting,  in  stratified  beds  by  folding  also.  The  movements  in  this 
zone  of  fracture  and  in  the  transitional  zone  of  combined  fracture 
and  flow  may  be  regarded  as  merely  the  responses  in  a  thin,  brittle, 
and  relatively  weak  outer  layer  to  deformative  movements  pro- 
gressing in  the  great  thickness  of  the  lithosphere  below.  But  the 
rocks  of  deeper  origin  which  have  been  exposed  at  the  surface  by 
profound  erosion  show  that  they  have  yielded  in  another  fashion. 
Their  foliated  structures  and  crystalline  textures  testify  to  yielding 
by  massive  flowage.  Fracturing  appears  to  have  been  absent, 
except  in  so  far  as  it  was  produced  by  intrusions  from  below,  giving 
rise  to  complexes  of  dikes  and  sheets.  These  visible  exposures 
suggest  that  at  still  greater  depths,  notwithstanding  the  great 
strength  of  that  zone,  open  fracture  planes  disappear  and  rock 
flowage  both  by  granulation  and  by  recrystallization  is  still  more 
distinctive.  This  appears  then  to  be  the  mode  of  yielding  of  the 
great  body  of  the  Mthosphere. 

Recently  Becker  has  suggested  that  fracturing  may  enter  into 
the  problem  of  isostasy  in  the  following  way:  The  demonstrated 
capacity  of  small  cavities  to  remain  open  under  great  pressures 
may  permit  fissuring  and  jointing  to  extend  deeper  into  the  crust 
than  had  been  previously  thought  possible.  To  the  degree  to 
which  fractures  and  porosities  do  exist  they  must  decrease  the 
specific  gravity  of  rocks.  If  shattering  pervaded  the  rocks  of  one 
region  and  not  another,  even  though  the  rocks  were  exactly  alike 
in  composition,  the  densities  would  become  different.  To  give 
isostatic  equilibrium  the  region  of  shattered  rocks  would  have  to 
stand  higher  than  the  other.  This  would  be  the  initial  effect  as 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  513 

a  result  of  the  decrease  in  density,  even  if  the  zone  of  compensation 
rested  on  an  unyielding  base.1  The  logical  correctness  of  this  argu- 
ment is  not  to  be  questioned,  but  rather  the  degree  of  its  appli- 
cation. The  following  arguments  suggest  that  shattering  or 
porosity  are,  however,  very  subordinate  rather  than  determining 
factors  in  the  isostatic  problem. 

Such  a  theory  does  not  account  readily  for  the  movements 
needed  to  maintain  isostasy  because  of  erosion  and  sedimentation. 
These  surface  changes  of  mass  suggest  a  restoration  of  mass  by 
lateral  undertow.  Furthermore,  the  appeal  to  nature  shows  that 
the  rocks,  once  deep-seated,  which  have  become  revealed  at  the 
surface  by  erosion,  are  almost  without  pore  space.  The  average 
porosity  according  to  Fuller  is  o .  2  per  cent,  but  the  mean  differ- 
ences in  densities  between  ocean  and  continent  which  must  be 
accounted  for  under  the  hypothesis  of  uniform  compensation  to 
a  depth  of  122  km.  amount  to  about  4  per  cent. 

Joints  are  observed  to  decrease  with  depth,  becoming  tighter 
and  more  distantly  spaced,  and  the  indications  given  by  the  lack  of 
general  circulation  of  ground-water  through  crystalline  .rocks, 
except  within  joint  spaces  near  the  surface,  are  that  at  greater 
depth  the  joint  spaces  are  negligible. 

In  the  great  compressive  movements  the  whole  thickness  of  the 
crust  must  yield,  but  even  this  cannot  be  conceived  as  producing 
porosity  by  granulation  sufficient  to  notably  modify  the  density. 
A  large  part  of  the  deformation  in  the  deeper  crust  must  be  by  a 
process  of  recrystallization.  Assume,  however,  that  granulation 
is  the  dominant  process.  Observation  of  granulated  rocks  shows 
a  reduction  in  size  of  the  crystals,  but  these  broken  fragments  fit 
against  each  other  perfectly  and  without  great  internal  distortion 
of  crystals.  In  granulated  rocks  from  the  zone  of  flow  there  is 
therefore  always  some  amount  of  recrystallization,  sufficient  to 
eliminate  that  porosity  connected  with  minute  shattering  and 
movement  of  the  broken  particles.  The  explanation  appears  to  be 
as  follows:  The  minute  shattering  of  the  minerals  tends  to  give 
a  high  pore  space,  but  with  a  high  pore  space  the  amount  of  contact 

*G.  T.  Becker,  "Isostasy  and  Radioactivity,"  Science,  XLI  (1915),  157-60; 
"On  the  Earth  Considered  as  a  Heat  Engine,"  Proc.  Nat.  Acad.  Sci.,  I  (1915),  81-86. 


514  JOSEPH  BARRELL 

between  grains  becomes  proportionately  less.  For  the  prevention 
of  ready  recrystallization  and  the  maintenance  of  this  pore  space 
the  granulated  rock,  according  to  present  theory,  must  be  con- 
ceived of  as  dry  and  the  grains  accordingly  unsupported  except 
at  the  points  of  contact.  The  shear  strains  within  each  grain 
become  very  great  in  proportion  to  the  diminution  of  contact,  and 
increase  in  proportion  to  the  regional  pressure.  If  the  points  of 
contact,  for  example,  cover  only  one-fourth  of  the  surface,  the 
compression  on  those  points  would  be  four  times  as  great  per  unit 
of  surface  as  if  there  were  continuous  contact  between  grains.  On 
the  intervening  parts  of  the  surface  there  would  be  no  pressure 
Internal  shears  would  result  in  this  way  from  the  hydrostatic 
pressure  of  dry  rock  due  to  depth  and  are  not  dependent  upon 
a  pressure-difference  in  the  rock  as  a  whole.  The  internal  strains 
would  tend  to  produce  molecular  changes  of  state  as  in  the  plastic 
flow  of  metals.  There  would  be  melting  to  relieve  the  strain,  and 
refreezing  by  which  the  molecules  would  build  out  the  crystals 
into  the  pore  spaces.  By  this  means  recrystallization  can  go  on 
without  the  aid  of  crystallizers,  though  presumably  with  more 
difficulty,  and  the  comminuted  crystals  come  to  fit  compactly 
as  they  are  observed  to  do.  This  elimination  of  porosity  pre- 
sumably goes  on  approximately  with  the  process  of  granulation, 
though  it  may  lag  somewhat.  It  would  go  forward  more  effectively 
with  depth,  irrespective  of  temperature,  since  there  would  be  the 
greater  static  load  upon  the  rock  and  the  greater  differential 
pressures  within  the  mineral  particles.  It  might  be  expected  that 
such  reduction  of  pore  space  would  go  forward  to  a  limited  extent 
only,  leaving  a  residual  porosity.  Observation,  however,  shows 
that  the  pore  space  has  been  almost  completely  eliminated. 
Furthermore,  the  rocks  now  exposed  at-  the  surface  acquired  their 
absence  of  pore  space  at  depths  of  only  a  few  miles  from  the  surface. 
At  depths  measured  in  tens  of  miles  there  seems  then  no  expectation 
that  density  would  be  notably  decreased  because  of  a  development 
of  porosity. 

To  sum  up  the  modes  of  yielding  within  the  lithosphere:  at 
the  surface  is  seen  to  exist  a  thin  outer  crust  intimately  cracked 
on  the  outside  by  closely  spaced  parallel  joint  systems.  Local 


THE  STRENGTH  OF  THE  EARTH'S  CRUST  515 

extreme  deformation  is  by  faults  and  folds.  With  increasing  depth 
and  strength  the  joints  become  less  abundant  and  faults  pass  into 
flexures.  The  passage  of  fractures  into  flexures  implies  the  begin- 
nings of  massive  flow.  Where  magma  tic  heat  or  emanations  are 
not  present  the  mode  of  mashing  is  presumably  more  especially 
by  granulation.  With  still  greater  depth  the  yielding  becomes 
more  uniformly  distributed  throughout  the  rock  mass.  Both 
because  of  this  pervasiveness  of  mashing  and  the  great  strength 
of  this  zone,  deformation  here  requires  the  most  force  and  absorbs 
the  most  energy  of  any  part  of  the  lithosphere.  At  greater  depths 
the  rock  is  more  compressed,  and  is  still  more  rigid  than  above, 
but  the  temperature  here  approaches  fusion;  recrystallization 
readily  takes  place,  the  strain  which  can  be  elastically  carried  is  in 
consequence  low,  and  the  lithosphere  passes  gradually  into  the 
asthenosphere.  Where,  however,  magmas  rise  through  the  crust 
they  carry  with  them  the  environment  of  the  asthenosphere;  the 
lithosphere  becomes  locally  abnormally  heated  and  saturated 
with  magmatic  emanations.  Recrystallization  goes  forward  readily 
and  the  zone  of  weakness  penetrates  upward  even  to  the  zone  of 
fracture.  Thus  in  the  injected  and  crystallized  roofs  of  ancient 
batholiths,  laid  bare  by  profound  erosion,  we  may  perceive  the 
nearest  approach  to  dynamic  conditions  which  prevail  in  depths 
forever  hidden. 


14  DAY  USE 

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